The first BCM module didn't arrive, so I had to buy another. I swapped the EEPROM chip, which was very easy, and installed it on the car. However, that got me back to the starting point. The car starts again and appears to work ok, but dies after a while as usual. I suspect that the old BCM module had cracked solder joints (nothing visible though), and possibly other issues too, which got worse when I removed the module. I was very careful with force and ESD, so not sure what happened. Anyways, maybe this new module solves random door locking issues and other funny things.
I think that the one last thing I'll try is to replace the ECU. Used virginized ECUs are not extremely expensive, so I'll probably buy one of those. If that doesn't help, I probably won't invest more time or money into this car.