You appear to be over complicating the issue. A supervised solo flight is one where the student does not hold a valid licence and therefore may only fly under the supervision of a Flight Instructor. A simple matter of law and nothing to do with CAP 804, a guidance document.
In respect of the FI(R) they may supervise solo flights except:
FCL.910.FI FI — Restricted privileges
(b) While conducting training under supervision, in accordance with (a), the FI shall not have the privilege to authorise student pilots to conduct first solo flights and first solo cross-country flights.
So if its not the first such flight, they may supervise it, under supervision themselves of course.