It might have killed the fob, electronics don't like reverse polarity.Plus the battery is now dead.. would leading it in the wrong way, kill it?
I can't see how it's related to the core issue, unless a bad signal could somehow crash the kvm? And it resets itself some hours later?
I've never had 'batt low' on the dash and never thought to check.. plus with the other fob not working assumed it was not a battery issue (batt in the other fob is good)