Using your own aircraft for training should not be a problem for a good flying school. You are hiring the skills of the instructor - you are not cross-hiring them your aircraft so that they can hire it out to other students. As such there is no need to add your aircraft to the flying school's AOC, nor does the school need to see anything other than the current maintenance release.
Just make sure the instructor is experienced in what you want to learn, and has sufficient experience on the type of aircraft you own. I've made the mistake of hiring an instructor who had less experience on the aircraft type than me (and at the time, my hours on that type were low, very low).