As far as I know, a public use airport does not have the authority to legally preclude you from performing flight instruction, period dot. You got misled, probably because you went to the source of the rent-seeking itself, airport management.

You don't even need to have a physical facility to perform part 61 instruction. I earned my PPL under the instruction of a freelance CFI who had to sit down with me at the local FBO or at the on-site restaurant to get the pre-flight discussion accomplished. Nobody said boo. Go on and instruct to your heart's content, nobody is gonna mess with ya, and if they do, give them the finger and keep on flying cause they got no leg to stand on.