LOE for A level Private Candidate

Has anyone managed to get an LOE (Letter of Recognition and Equivalency) from the CDC as a A-Level private candidate?

The problem is that there is no school to issue a character certificate, since you sit exams through the British Council directly. The British Council can attest your Statement of Results and can provide a migration certificate but won't issue a character certificate.

If you found a workaround, how did you do it?