What type of microphone is best for a medium space room mix?

I know I could mix a room mic with a fitness mic (something the teacher wears), but I'm wondering if there is a suitable mic that would remove the need for two mics.

You want to capture the ambience. The class is the ambience and wants to hear the teacher. Case closed.

Since you say you are going to pick up the teacher via the PA, all you need to worry about is the ambience. You'd use a cardioid or hypercardioid condensor for that and put the teacher's PA as well as possible into its dead zone while pointing at a representative part of the group. Prioritising that group over the PA may require moving up a bit: try not to be creepy. If the distance is a bit more, a reasonably quiet large diaphragm condenser would help you avoid hiss (hiss sounds like electronics, not like ambience). You'll want a good microphone suspension (spider) and possibly a lowcut on the microphone to avoid stomping noises dominating the recording levels.

Whether you do some lo-fi and reverb treatment of the PA part of your recording to make it blend in better is up to your choice: a clean take from the instructor mic leaves open your options.

I've been avoiding answering this for the whole day, because my overwhelming thought is "you don't want to do it like that".
Ambience is going to be the enemy of clarity in that type of environment.

You're feeding a 'fitness mic' - which I assume is some kind of cheap headset mic near the mouth - into a PA in a very live room.
Any other mic you put in that room is going to be ambient - whether it's 3ft or 10ft from your instructor. You do not want to be feeding this mic to your PA at all.

The only solution I can see does involve both mics.

Your 'fitness mic' feeds to the PA & to your recording. Get the PA at the opposite end of the room from the instructor.
Your ambient mic feeds only to your recording. The further away from your instructor & the PA, the better. This is [attempting to be] purely for room sound, so you want to avoid the main voice as much as possible.
You judiciously balance one against the other using headphones before feeding it to your recorder, unless you are recording multi-track so you can mix later.

This will all need to be mono, else you will need two ambient mics.

As for mic choice; you may as well just go for a dynamic cardioid on a wall, pointing at the room centre.