I don't know if it's a real issue. Historically the Mongols often allowed native lords to rule their lands as long as they paid tribute and recognised overlordship of the khan. And there's plenty of examples when Russian princes revolted against the Golden Horde. Unless Aztecs bring their own administration and bureaucracy overseas then they must rely to vassal lords. I'm not expert in Mesoamerican history, but I believe that Aztec Empire made other tribes and city states to pay tribute to them instead of completely annexing them. So having them to rely on to foreign vassals in Europe doesn't sound that unrealistic.
But were not talking historically, but game mechanics.
For a ruling culture/religion to survive in power in game it needs to have atleast 1/3rd of the land/vassals of the realm. Or else it wont have enough of a foundation to fight off civil wars, plots or factions.
If the AI is winning the war on battles in after only 3 provinces have been occupied, then the invasion is over as most likely they will accept the first factions demands against them and even if they refuse it they have no chance of lasting when 9/10 vassals are hostile.
Thats how the game works. If youre powerless and all your vassals hate you, youre going to lose. And with how soon the Aztecs accept peace theyre always poweless and they never have enough friendly vassals to counter the hostile.
If you use the console to give_title's to the Aztecs in that intial kingdom they conquer so they have that power base to work from, then they actually pose a threat to their neighbours, they can keep their vassals under control and so theyre free to invade.