An old Megger or AVO?
As you can see this is a rare and bizarre fault that you have here. One that, apparently, a lot of people have not seen before. The BSIs tend to be reliable and most of the common faults have been sorted by now. So on one hand we are curious to solve it but on the other, we don't want you to waste your time or money sending the BSI off if it's serviceable.
I have been a technician for a very long time and I can tell you that nothing is guaranteed until the thing you are working on is fully functional. You can be dead certain about a diagnosis...replace the component...but the thing still doesn't ^%$#ing well work!
The first rule to fixing something is to understand how it works in the first place. As you can see we are floundering a bit here. There is the way you assume something works and then the way it actually works. I can't speak for any one else here but I didn't attend the PSA technician's course. So you will have to take what I say on face value.
Does the BSI repair company have fixed prices? One for testing, one for repair. Or a variable price based on the time to repair?
Have you checked out the price of a cheap BSI from Ebay or the breakers? Or are they just not cheap? (I don't live in the UK)