Following the problem with the used wing mirror I bought a new third party one. For some loony reason it didn’t come with a mirror glass, but fortunately I could use the old one. After a great deal of pain I managed to screw it on and re-fit the door panel and now I have an MoT.
I mentioned the concern other places had had about the exhaust fault to the MoT people and they were most dismissive. I think I will investigate other MoT providers and see if I can get a better deal.