>> the range extender was unable to recharge the battery

Could you expand on this? What was the actual problem? For example, did the range extender start and run? Did it put any energy into the battery at all?

We ran out of battery and fuel (I was on the backseat so I don't remember the exact sequence, I just remember the car basically losing power without much warning on the autobahn). The car was towed to a gas station were we proceeded to refill the tank, but the range extender would not start and the car computer was adamant we had to recharge. Which led to a quest to find a public charger in the middle of the night, after 2 failed attempts, we gave up and asked to be towed to our destination instead since it was close enough.

The original BMW i3 had/has a lockout that prevented the range extender from charging the battery until it reached 6% in order to meet California's ZEV BEVx regulation.