Looks like nobody is going to solve it at this rate. So here's what you should do first:
Retrieve a bunch of teleport bombs in the O area as well as a master key (or several if you want to play things extra safe). Yes, these things are still helpful (albeit not essential) even if you're not allowed to use them to unlock any of the save room doors. Teleport bomb yourself to the save room as well. This way you don't have to unlock the O door.
Unlock the A door with its corresponding key. Remember there are no restrictions on entering the main part of the village. Just make sure only to leave it by either teleporting or resetting. Find ways to retrieve what you need next without breaking the rules.
You need to get the L, H and P keys in the village. The E area is safe to skip because it contains no unique animals. And the trophy room can be entered through the village main area, though this will use up one of the two times you're allowed to open a door there. This means you are only allowed to open a door once in the process of retrieving the keys.
As mentioned in another topic, there are tricks that allow you to jump to extreme heights. Use that to your advantage.
If you want to get a teleport gun, there are two ways to do so in this challenge. There's a very small chance one will spawn in the village main area on the floating platform with the detector detector. If you don't have the luck or patience for that, there's another way, but I already gave you a big hint on how to get that one earlier.