That's a bloody awful situation to be in for all concerned. A decade ago I'd have said no US President would ever do something that stupid, but ... well, let's assume it happens.
Butler was convinced that the United States should have gone to war with the British in 1862, and that they would win. In this situation I found it almost impossible for Butler to pass up as it allows him to hit the British indirectly, or at least that is what he thinks initially. British intelligence would make it quite clear that the American government was all but openly supporting the Fenians.
The Fenians have a core of veterans, and they know the kind of warfare that characterised the last years of the ACW, so when they crash into Canada, it's going to be nasty, farms burnt, cities shelled, towns put to the torch. Atrocity stories like that are not good. Assuming they manage to defeat the unprepared British troops and occupy the St Lawrence, they won't be popular with the Anglo-Canadian population. I assume they'll try to ingratiate themselves with the Franco-Canadians as fellow Catholics under the English boot. Not sure how well that will go over, but I doubt the relationship will be as cordial as the Fenians may imagine. Their problem is immediately: what next? They are going to find it hard to rule Canada. Even if the USA recognises whatever state they'll declare, nobody else will. They are facing an unhappy population, the damage they did will hamper their ability to realise revenue, and they need a lot of revenue because they have to keep their militia functioning and prepare for the counterattack they know will come. I can't see the USA providing that much support even while they still can. So, how good are the Fenians at policing and tax collection? I'm guessing pretty awful.
The Fenians that did make it into Canada did not undertake those kind of tactics, though that was with a far smaller number and I can easily see some excesses. However something akin to Sherman's March I cannot see as they would need that infrastructure to properly occupy the area.
I can see them working well with the Franco-Canadians, at least initially; it will be far from universal, but if it means that they get to control their own affairs for a time (at least if they don't interfere with the Fenians), then I don't see much going wrong in the initial year.
The United States will not recognize any Fenian state as the Fenians never would have established a state, at least the kind your thinking of. At best they would have appointed an Irish "Government in Exile", offering Canada in return for Irish Independence or, in the view of many, Home Rule. Butler might want to recognize them but the United States Congress would have more sense than that.
The Fenians were largely paid for by donations from Irish-Americans. I don't see that changing, though the collection of revenues would allow for additional expenditures where needed if they decide to go that route.
Butler has his 'mission accomplished' moment and then realises he now needs to go to war with Britain. The absence of food imports from the USA and canada will be a problem for the British, but not an insurmountable one. Higher food prices will spread out over Europe for a few years as grain is bought up from other markets. Nobody in Britain will be moved to demand Canada be given up so that they can eat, though - this will be viewed as strictly the Americans' fault. His problem is that in facing the British, his bargaining chip - 'we'll invade Canada' - is gone because his unreliable allies have already done that, and he can't hope to defeat the Royal Navy. The Union Navy was an impressive establishment, but it wasn't at the same level. He won't find many allies internationally after sponsoring a nationalist insurgency, and he still has to face up to his country's internal problems. How eager will the young men of the Union be to join the colours and fight the British for the freedom of Ireland? How eager will the former Confederates be to fight anyone but the Union states? How is he going to pay for the armies he is now going to need, with his massive debt, inflated currency and war-weary population?
In some cases I have read as much as a third (~30%) of all the United Kingdom's food ("grains") came from the United States, so I'm not certain how dire the situation might become on that score even were they to rectify that somewhat. By extension there would be severe economic losses as exports to the United States would cease which was, I believe, the major market for their manufactured goods. Again not sure how that might effect things as I can't really determine the size of this economic loss without proper numbers.
The Fenians would have only invaded the populated strip of Canada along the St. Lawrence River Valley and what was then known as "Canada West", basically the area from the River Valley to Michigan. That would have left much of Lower Canada open to American occupation as well as New Brunswick, Nova Scotia, Labrador, Newfoundland, British Columbia, Assinobia, Manitoba, etc. etc. Except in the case of Newfoundland there would be little to oppose an American invasion. Instead of part of Lower and Upper Canada to bargain with, Butler now has the whole of British North America (well, outside the Caribbean region).
The American people will be tired of war no doubt, but if the United Kingdom simply declares war on them, even if the reasons are later publicly stated, they are liable to support it for a time. It helps that general feeling towards the United Kingdom was at a low given their controversial support towards the Confederacy, at least as it appeared. The South is liable to be done with the fighting entirely, though I can see some hardliners hoping to start up a new rebellion; at this time, given Congress and Butler I could see passing the 13th and 14th amendments (the latter of which makes the regulation of elections a Federal matter), I saw the rise of the KKK not as a terrorist organization but as an insurgency, in effect the hardline faction I mentioned. They decline over the next couple years, unable to effectively combat the Federal occupation, but they might find some new recruits in this situation. Support on the part of the British however I don't see as clearly, or at least on a level to matter.
Butler was a major supporter of Greenbacks and wished to make it a permanent institution of our currency, so I don't see inflation as much of a concern unless it becomes problematic even with that instance. The Revenue Act of 1864, which had been the final one to set Income Taxes, I can see being renewed in 1866 in light of the situation in combination with the support of Butler even before the Fenians moved into Canada. If that would make a dent in the expenditures under which the Federal government would operate I'm not sure, but it would be a start.
Assuming he can muster a large enough force, what does he do? He could march into Canada and take over from the Fenians. How will they like that? What good does it do him to sit on an occupied territory, and how will he come to an agreement with the people who gave it to him? I don't think this would be wise. Alternatively, he could try to send the US Navy into the British Caribbean in the hope of preempting an enemy build-up. Maybe land troops in Jamaica and on the Bermudas. The problem is he would need to do that quickly, and doing anything quickly is not what the US government is good at. But if he could pull that off, he'd seriously have inconvenienced the British who now have to fight a naval war against him without their bases to the north and south. If. IMO the more likely outcome is that the British have their pieces in place earlier and the expedition either is canceled or fails.
A lot of this I already made mention to above.
The naval bases in Canada I can see being taken by the United States relatively easily except in the case of Newfoundland, but given its importance as a Naval base for the British historically, I don't see the United States dithering on whether to make any operations against it in contrast to the Caribbean.
In the case of the Caribbean there is only Bermuda, which was the preeminent Naval Facility not only in the Caribbean but in the Western Atlantic. This in my opinion is a toss-up if the Americans manage to reach it first, as the position is relatively well developed and fortified, advantages the United States can treble with more men at their disposal not to mention distances. The British however can also reinforce their positions there, or defeat an American invasion force that proves insufficient, and make the position one where a first or second expedition is untenable.
The British are not going to be divided on the war. It would be like expecting the USA nmot to react too forcefully to Pearl Harbour because the Republicans in Congress oppose interventionism. So they'll start pretty much immediately to hurt American commerce, very badly. First, they'll hoover up any US shipping still abroad. Then, they'll start a far blockade of US ports. If the US Navy comes out to fight, the result is most likely an American defeat (if they are incredibly lucky and beat the Royal Navy, the USA might pull off a victory by points, but I doubt it very much). To the Americans, the more appealing strategy will be to have their coastal monitors distributed widely to mount an effective defense against close blockade than putting all their eggs into one basket. Most of their warships will be badly outclassed anyway. This way, the British can't shell ports and land troops at will and at least need to exert themselves.
Agreed with much of the above, though it may strain relations with Europe.
How badly do the former Confederates want to see the federals hurt? What reception are British troops going to get in Mobile, New Orleans, or Galveston? IIRC Butler was not big on reconciliation. And then there is Mexico. A nice long border and a manpower reservoir that, if mobilised against the United States, could deliver some nasty stings. Or they could just assemble a large enough fleet, bull through the US Navy's defenses, and start burning port cities.
I don't see the British doing more than raids as you outlined, though even that I'm not sure depending on the nature of the defense both at land and at sea.
Mexico is a non-issue. Butler followed a policy very similar to Johnson and forced the French out of Mexico; the only difference is that I saw him actually send a small contingent that would capture Maximillian, who would survive and be sent into exile in Europe. And no that didn't mean he was fighting the French; the contingent was only sent after the French had agreed to withdraw, but it became clear that the Maximillian loyalists intended to fight on.
Confederates I've already mentioned.
Eventually, the British will want to land a real army. I'm assuming that by that time, the US political and economic warmarking capacity will already have been badly degraded. If not, that could backfire for them. They can't match US numbers and the quality of troops is higher than it would have been before the civil war. But realistically, how many more men than they need for coastal defense and sitting on the south can the USA put into uniform without risking draft riots and massive desertion? I think that is really their Achilles' heel - they need the support of the people for a large-scale war like that, both to fight it and to pay for it. I can't see President Butler getting it, certainly not into the second year of the war. The British, OTOH, can command the resources and raise the funds they neede, their credit undamaged by an exhausting war and the navy already existing.
I'm not certain how badly the United States would be damaged economically, as many manufacturers for example would for the time no longer have to compete with their British counterparts. Agriculture would take a hit, but then it could also potentially be shipped through Mexico provided such a route is available (and I have my doubts). I actually think that the British would suffer more damage economically, as they exported a significant amount to the United States while also importing a large portion of their grains from them, as outlined above. Of course as you mentioned the British are starting from a far better position initially.
United States Navy would still exist at its Civil War strength, the downsizing not happening until 1867.
I can't really see the same kind of desertions and riots against service that were witnessed in the Civil War, in large part because there isn't going to be any extensive land combat except in those cases where the British attempt to land an invasion or raiding force. It would certainly still occur, but not at the levels as were experienced during the Civil War itself. African Americans from the South could also be drawn upon as manpower for the Army and Navy, though that might inflame tensions in the South.
Support is key, I agree, though I think the United States could manage keeping its revenues at a level where they could "afford" the conflict. With the conflict deviating away from Ireland and more towards the conquest of Canada as it would seem, I don't see the Irish-Americans being supportive of the effort as they would have earlier. The people in general would be weary of war. Then again, until the British are actually landing men on American shores the war would be almost purely naval outside of the occupation of Canada, which does not convey the same kind of emotions that for example Cold Harbor did. Maybe a grudging support, where the population now simply wants it to end, but on American terms or in part.
