I'm 99% certain that reversing lights aren't part of the MoT. I certainly can't see it in here:

MOT Testers Manual

Chances are it's the switch on the 'box though.

Number plate lights, if it was the switch they would all be out. If the plate is in the tailgate, chances are it's either a broken wire somewhere. However, if it uses spring load pins and copper contacts like my old Escort did, have a look there. as they can get broken if the tailgate gets shut with anything trapped.

The thing is £320 + VAT sounds like a lot to spend on a worthless car but your chances of finding another car for that amount of money that doesn't need work are pretty slim. Shopping for cheapo cars can be hard work because of the sheer volume of dross out there and the time you spend looking could be spent sorting out some of the problems you've got.