Most schools I've seen do open-enrollment, set classroom courses in the evening. If you offer personalized, intensive, one-on-one groundschool in the daytime for a higher price it might not even compete with what the school offers. Instead, it would complement their service.
Your success would also depend on the enthousiasm of the FIs to conduct groundschool classes. Some like it, some don't. A passionate GA instructor might like to do groundschool, while hourbuilding-airline-wannabees might not be so keen.
And you could even consider some sort of kickback/referral fee for the school who brought you a student, to soften the emotions somewhat. After all, a (commercial) school makes money on the FIs it employs. Why should the school not make money when they refer someone to an instructor who happens not to be employed by the school?
So as I said, do some investigation first and consider what sort of deal the school and its students would be interested in, before stepping on sensitive toes.