TOO,
It is not about how Instructors get paid (employed, self-employed, etc) it is about who provides the service.
"...get students/candidates to hire the aircraft from the Club, then enter into a separate, private arrangement with the instructor"
Perfectly acceptable, I would have thought, for Examiners and Instructors for continuation training but, for Initial/Renewal training, you cannot 'Have your cake and eat it'.
Either:
a) The Club is providing both airctraft and instruction and needs to be an RF/ATO
or
b) The Club is only renting aircraft to members and the Instructor is providing instruction, in which case they will need to be registered as an RF/ATO.
From what you have written it sounds as if a) above applies but, for solely Tax Avoidance reasons, you are charging as if b) applied. Be careful !