There might be a bigger problem. The 40 amp reverse polarity fuses should have blown.
Since you still have 13v coming from the convertor they didn't. This could have ended with the battery exploding. So check and see if the fuses are there and not replaced with a "no blow" wire.
All your 12v systems will run off the convertor as long as your connected to shore power.
I can't say how anything worked if you weren't.
All this based on the black lead being negative in this installation, which it isn't in some cases
It'll be interesting to see what others think...