Using your own plane shouldn't be an issue, but a couple of points to consider:
1) no need to list specific plane or type of plane on a school's AOC
2) nevertheless, a lot of schools don't like this and will not be enthusiatic about using your plane. They will probably charge you a higher hourly rate for the instructor and might even fob you off if they are busy.
Small flying schools outside the metro are are more likely to be flexible. I've been doing some training at a small regional school and thinks it's great.
3) on condition shouldn't matter, just need a current maintenance release.
I've used my plane for training, and apart from being familiar with it, another benefit is the hourly rate for my plane probably costs me half what it would cost from a flying school.