Converter/battery

A one volt drop between the converter and the battery is not "as good as it's going to get." There's a bad or open connection somewhere between the two, or a seriously undersized wire for the charge current (and, at 12.7V, the charge current is not likely to be very much anyhow).

I'd double-check the battery cables, including the ground cable, looking for corrosion or other problems at either end. I've seen a couple battery cables that looked okay at first glance but had the copper wire nearly or completely corroded away internally.