Hi Mark,
I dont know if you have gotten this sorted by now or not but just to add my own experience to this as it may help someone. I had a similar issue myself on my F11, were there was a drone from the rear at speeds of around 50mph. I also thought this was coming from the diff, and as Greydog said, the changing of one is about three hours work and can be expensive, even if you do it yourself. Ive done it on a E90 3 series in the past and didnt fancy doing it again on the ground.
I did check it myself for all the usual suspects, including the rear wheel bearings which showed no signs of movement. I crossed my fingers and changed the diff fluid and added an additive from LIQUI MOLY to it but sadly this didnt improve the noise. I had suspected it was actually the rear bearings as the fluid taken from the diff was only 12 months old and changed by myself and when I did the latest change the fluid wasnt grey and didnt appear to have any particles in it. So at this point I decided to change the rear bearings anyway as they were probably in since new (2013) and they have a tendency to fail anyway and this car is getting all rubber bits and known common faults changed over time anyway.
What i had discovered was that the rear bearings were seized into position because they had failed some time ago but didnt make enough noise to be noticed. They were so badly seized that I couldnt get them off so I sent the car to a local garage that I know well and he also couldnt get them off. In the end I found a specialist with a really big hydraulic pullers (normally used on VW transporter front bears apparently) who made light work of the job. Did this get rid of the groaning coming from the rear of my car...... yes it did! It is back to being a nice quite cruiser again.
Im not suggesting you are wrong about your diagnosis of your car but im more suggesting that these F10/11's are very well insulated from the road noise and this stopped me from hearing the sounds of the rear bearings failing over time which lead to them being welded in position. This meant that when the failing bearing finally made enough noise for me to hear it, it was sudden enough to make me think it was the diff instead. Also the rear bearing didnt have any noticeable wiggle which would typically be an indicator of imminent failure.
Diagnosing the difference between these two possible failures can be tricky at times and both can be expensive to repair, so for that reason I'd like to say to be sure of what the problem actually is by maybe asking a diff and gearbox specialist to have a look also. Because If had changed my diff out and found it was actually the rear bearings I'd be kicking myself more than a pro goalkeeper at a free pedo kicking competition.
Hope you got you car sorted and for a reasonable price too. Happy motoring.