I've had problems with this aswell, car failed 2 times on a rear wheel bearing before I finally got it to pass. I thought I was doing the nut tight enough at home and I would test for play with the car jacked up but on the drive to the MOT station it would still develop some play which i didnt have before
your old school mechanic is correct that the nut shouldnt be too tight but he 's being too careful with it, they still have to be 'sort of' tight
also because its a new bearing I can imagine it has to wear in a bit and needs re-tightening after a short while
if the nut is tightened too much the wheel will rotate less smooth while the car is jacked up. try to tighten the nut atleast 1/8 or 1/4 of a turn more than you would think is right and see if it still turns smoothly (car jacked up) and let it be tested again
otherwise just ask your MOT guys to fix it as it is hardly more than 15-30 mins work to tighten the nut a bit more.