[EricH]

Start with this: Go to the administration. Inform them of the situation.
1) You are not allowed to use the FFA shop due to the teacher not wanting you to use it.
2) The one teacher who likes FIRST says you're making a mess and kept key robot parts (the batteries are critical) away from you. You'd like a mediator to discuss this sort of thing.
3) Your teachers don't seem to be very interested (aside from one or two); you would rather have interested teachers than "I have to be here because administration wants me here" ones.
4) You have limited build time due to when teachers are there. You'd like to know if other school staff would be all right if a teacher can't be there (trust me, you DO want to know this--it can expand your work time a bit).

If the administration cannot help you with the teacher difficulties, then you want to look at going independent. With strong enough support, you don't have to fall.

You have multiple options there. 4-H is one; a 501(c)(3) is another. I might recommend going through 4-H; talk to 1902 or 1501 on that. They may also know more about places to meet and things like that.