After we went through everything again exactly and could exclude some other parts (the error was still there in the Limp Home mode) it was the same error as with my car. Stuart completely bypassed the BPS switch, even though it was new, and the car has been running ever since. You have to push the button of the switch very far in for a permanent good contact, of course without overdoing it. Because I didn't trust the switch or its location, Stuart bypassed it with the good and now permanent result.
Thank you - I obviously didn't read far enough back in the thread.
