On the other hand, check right at the output terminals of the converter to see if you get 12.6 there. It could quite likely be a loose connection in the wiring to the battery.
I had mine plugged in to shore power and thought everything was OK. We would park for a couple of days plugged in, then drive for a few hours. Finally we stopped for three days and I watched the idiot light and the battery power get lower and lower while plugged in. When I put my meter on the output terminals it was putting out full voltage. I found the white neutral wire on the converter terminals was loose. Tightened it up, and it worked fine.