Well here's a new one...
Phone/MFT were working fine last week - playing together just as you'd expect. Got into my car to drive to work and BT wouldn't connect. Usually a reboot of the phone fixes this (it's two years old and can be slow at times) but not this time. Rebooted again, just to see, and no dice.
Tried turning BT on/off in the MFT settings menu, deleting the phone, re-pairing. Wouldn't work.
Interestingly enough, however, the pairing request I saw on my phone was totally screwed up. Instead of the normal "Sync" device name, it was trying to pair with the BT chip itself - or something like that?
Eventually I got fed up and hoped the system would reboot itself. A day or two later, still not working, I finally pulled the fuses. MFT "performed maintenance" and I was able to re-pair with Sync like normal.
It's been fine since.