I think its fixed.

I changed all ribbon cables with no effect on my problem, so I called MCE again. The troubleshooter had me check for a loss of communication on the swing panel of the group controller, turns out I was in fact losing communication.

MCE has two different types of "receiver-driver" chips on their communication (RS?) boards. Looks like the problem car and the dispatcher have the weak version of this chip and my nice elevator has the good version of this chip.

To eliminate communication loss, there are two jumpers on each communication board; they take a resistor to ground at the end of the communication link. I moved the jumpers on both cars, my communication is now good and my problem disappeared.

So, I ordered new chips for both cars (a few bucks a piece).