I think the biggest problem is definitely foreign aid, and the power of the Inner Mongolian nobility, not military strength of the Mongolians. In OTL the Mongolian invasion of Inner Mongolia was actually shockingly successful despite being outnumbered and outgunned. The problem was, the nobility were mostly loyal to the Qing, and in Eastern Inner Mongolia, they were relatively loyal to the RoC, and none of the nobles really approved of Outer Mongolia's government, then, because Russia didn't endorse of the invasion, they refused to supply the Mongolians, which put all responsibility for food and supplies on the local Inner Mongolian volunteer troops and the Mongolian government. As far as I understand, the problems were mostly the financial issues of lack of international support bogging down the invasion and causing local volunteers and civilians to loose confidence, as well as the nobles being brutally repressed by the Bogd Khan's government.
wait until after 1916 to be able to do so
Problem with a post 1915 invasion is that the Mongolian government were sorta forced to renounce Pan Mongolism after the Kyakhta Treaty, which, itself isn't that serious because of both China and Mongolia mostly ingoring the treaty when Russia was no longer in the picture, but because the Pan Mongolists lost support in government, for example, Bavuujav, who defected to join the Qing Royalists and became a bandit.
after getting invaded by the USSR its chances to get Greater Mongolia are slim
Pretty much, although, important to note, "invasion" isn't the term I'd use for the soviet intervention as it was more of a military support operation called for by Mongolian socialists to free them from the Chinese and White Russian occupiers. Also, there were some notable attempts at Inner Mongolian independence post 1924 which should probably be mentioned, the left wing faction of the Inner Mongolian People's Revolutionary Party, through the Comintern renounced it and the Mongolian government arrested them after, did organise some rebellions in Hulunbuir with promised, though never delivered, Comintern support.
Whites win the RCW and support Mongolia in their expansionist policies
The only White's who really cared about Mongolian expansion, that I'm aware of, only did so in an attempt to win over Mongolians as a puppet or colony, and in the process of that, people like Sternberg did horrific damage to the nation and its people.
quite monarchic at the time and the nobility is weakened due to Chinese collapse and if they are opposed to the new government they would oppose their invasion meaning a convenient way of getting rid of them
Well, the Mongolian government did try this during the invasion, arresting nobles, forcing them to submit under duress and appointing alternate nobles who would support them, this didn't really work, as demonstrated by the fact the invasion failed. The problem is forcing the nobles to submit is that, if the invasion succeeded, they just gained dozens more nobles, and every spot of land in the nation, other than a couple square miles of land is owned by nobles. The theocracy was also extremely incompetent and levied extremely high taxes against the people and nobles which, as you can guess, no one liked.
