I have beaten the baba is you puzzle game, but I don't know much about Timmy.
Baba doesn't have a leading move but has a ton of options. How long it takes to execute one of those options basically depends of the position of text, so Baba will likely do what is most convenient first. It also seems like Baba will be able to see all of Timmy's properties simply by looking around, which can then influence his decision.
I don't see a lot of resistances on Timmy's profile, but even so, Baba can break those resistances. For instance, Timmy's fire resistance can be nulled by breaking the "Timmy IS NOT MELT" sentence and instead forming "Timmy IS MELT AND HOT". So it's more about his ability to come back from being slain or incapped by one of Baba's moves.
So can Timmy come back from being: possessed, transmuted into a rock, transmuted into baba, sleeped, erased, power nulled or melted? Worth noting that in the game, the melt property leaves nothing behind unless stated otherwise, so it's more like vaporisation or further. The other thing that's worth noting is that Baba doesn't need to keep reforming the rule to apply its effects, so if Timmy were to wake up without breaking the rule putting him to sleep, he would immediately fall asleep again, unless granted a resistance to sleep (which baba can then break, assuming absolutely everything exists as text).
The other thing Baba might do is take control of more things with the YOU text, so that simply destroying Baba's body isn't enough to actually stop him. If not stopped, Baba will be able to manipulate multiple rules at once and keep Timmy in check that way before eventually forming a huge elaborate set up that'll do terrifying things.
Baba has ways to target multiple things at once, (via GROUP, ALL and NOT+BABA) so he doesn't just have to target Timmy, he could target Timmy, the wands and the fairies all at once with the right words.
The way to beat Baba in my mind is to survive at least one of Baba's hax by coming back from it quickly enough, before Baba does anything else, then stopping Baba from manipulating any more text... Unless Timmy is fast and smart enough to stop Baba from doing anything right away, even though Baba's first move might be to break Timmy's properties.
I vote Baba for now.