The reality is even with a poor converter, two batteries should have taken you through the night after a 6 hour run of the generator.
Maybe, maybe not IMO. The different battery manufacturers have different specs. One I pulled up gives an acceptance voltage of 14.4 at 70 deg and 15.2 at 30 deg for FLA batteries, a difference of 0.8 volts. If the converter is putting out 13.6 volts, that would be the same as trying to charge the battery at 12.8 volts at 70 deg, or in other words, it's not going to charge worth a hoot. This is one case where the sorry direct-from-generator charger with its high peak voltage, unfiltered output would work better. As many of us have learned, without an ammeter to monitor, you really don't know what's going on.