I think you have a good reason to be sceptical.
Without opening up the diff there's not that much to say about the condition, so changing without proper diagnose is tbh not a really professional way to do things.
Do you know what they exactly ment by play? Have you seen it?
Was it axial play? radial play?
How much oil was left in the diff?
The play can be caused by a lot of things, like the flange nut not properly tightened. True that can cause damage to the bearings (those can be changed) or gears, but without opening up the diff thats rather a guessing game. It could be that you only need a new claping bush (thats a £4 part on its own) or that any play has already damaged your pinion/ringgear.
I dont know any UK based bmw diff specialists, but its sure time to get another specialist than the one you have now.
I'm sure there are some UK based companies that specialize in bmw diff refurbishments. Getting the diff there might be a pain because of shipping weight and the possibility that it can leak oil (some parcelservices specifically note that shipping car parts that contain oil is forbidden)
Usually a total refurb costs between 500 and 1k (thats usually cleaning/painting/sandblasting, adjusting/setting tolerances, new seals and (partial?) new bearings). An M diff will be at the higher end of that scale. And if new parts like pinion/crownwheel etc are required that will be more of course.
For these kinds of jobs you musnt go to 'bmw specialists' that do maintenance on cars but to specific driveline/transmission/diff specialists or motorsport oriented shops as fiddeling on a bmw diff is beyond that what is written in the BMW factory workshop manuals (so a 'bmw specialist' or a bmw dealer wouldnt know how to proceed). Its specialized work.
A quaife lsd wont solve this problem as the quaife diff is only the internal diff core. These problems are either related to the bearings/pretention or the crown/pinion.