Does the AI "cheat-peek" when selecting which Pokémon to send out?

After some research I've looked and from my limited knowledge I can say it's likely just you.

Based on some sleuthing Pokemon Stadium (and likely 2) are based on Pokemon Red / Green which after looking at some of the source code for Pokemon Red and searching online for reports I don't think it peeks at your party to calculate it's next move.

From the limited source code for Stadium I can also say there's no peeking that I could see here. This is all very basic looking and possibly not a direct source. Satoru Iwata did port the code based on the linked interview and could have possibly changed it in Stadium and Stadium 2 however.

This question isn't answerable via traditional means, but from my personal experience with programming and other Nintendo games, I'd say likely the opponent's pokemon isn't based on your pokemon specifically, but rather it's based on your inputs leading up to the fight.

try picking the same pokemon twice (even though it won't let you) before you pick the rest of your party, compare that to the normal where you pick your pokemon once from a savestate, if it's different it's based on your input if it's the same it might be based on your party?

Sorry, this is a little late, hope it helps, this is the best test I could think of for this.