Did a bit of searching, not much comes up on that, a few on K1200LTs with broken cats that had to be replaced, one also where thay had to replace the Motronic brain to cure that, but he may have had other issues too contributing to the failure of the brain.
However, you are giving me a clue, re exchanging them exhausts....running with "straight pipes" may have cooked the O2 sensor, that would certainly cause all sorts of mixture problems. Looking at the parts fiche, if that bike has a cat, it has to have a O2 sensor.
Your neighbor ran straight pipes.....did he also ran leaded race gas in there....! Ooops...not good for Cats and O2 sensors!
So...question here, was the O2 sensor removed when converted to the straight pipes?
If not then the brain may have learned some bad habits....that may take some miles to clear out, and that is after installing a new O2 sensor and resetting the TPS.
That BMW sensor is mucho $$$$, but can be replaced with a Bosch Universal at a fraction of the cost.($60.00)That would be Bosch 15730, you will have to splice it in there but that is easy, and a very long thread on this site on that one.
But here is a test you can do to see if yours is doing what it should:
But....if everything got as hot as you said, I think the sensor is cooked/toast!
I have always been suspicious of used electric/electronic parts, if they have any wires attached to them I stay away, most of them are hard to test so used parts dealers give exchange warranties, but there is always the odd shyster trying to recoup his investment in new parts....
P.S. now he has twins....he is getting back what you guys got from the "neighbor with straight pipes"....I've been there!