His fins were damaged, so keep the water level shallower than usual - easiest if he's in a smaller quarantine tank - so that it's easier for him to pop up to the top to breathe.
Keep the water clean. Keep it warm. And yes, the washed, dried oak leaves can put soothing tannins in the water. If you don't have any IAL, you probably won't be able to get any in a quick time frame, but Oak is usually readily available.
AQ salt is one of the cheapest cures you can buy. Lots of bang for the buck. It's seen me through lost scales, fin rot, and is now in use on my little boy's betta that looks like it might have gotten a bacterial infection on his fins.
Good luck, I hope he makes it. And definitely look into baffling that intake - you can find aquarium grade foam/sponge in pet stores.