I real long shot but are you not getting power when the stereo is ‘loose’ or not in its cage?
the only reason I ask is the Alpine halo9 I put in my wife’s Touareg (and murano before that) doesn’t power up until its touching the cage itself, (I.E earthing itself) never had a stereo do that before and it’s odd it does it in 2 different make of car but may be worth a look
Other than that have you got the red/yellow leads the right way round on the harness to allow it to switch and save its presets etc
final thought, a question to everyone this time, do you need some sort of canbus adapter on the later Minis, I had to add one to the Touareg as the OEM stereo wiring didn’t have a switch function, that adapter allows the stereo to turn on and off on the ignition as per the standard head unit, prior to fitting that I wired it to a spare fuse ‘hole’ but it meant it only came on with the engine running and went off instantly when the key was turned back to ‘ACC’ position