You're absolutely right that there is no legislative impediment to this and I was aware of the Arnold Baker reference, although I wonder whether it is intended to refer to the specific role of the Proper Officer. As we know, cases of a Councillor acting as unpaid Clerk for a short period during a vacancy are not unusual.
In this case, where a Clerk/RFO is in post, one wonders how the various roles can be dovetailed into an effective administration, without giving the Chairman undue influence over proceedings, which would, of course, be contrary to the basic principles of local council democracy.