I know, a full battery reads 12.8 open circuit but ...

With solar the voltage sits at 13.4 most of the day.

Before solar yes the voltage would drop about .1 or .2 under light load and I would run the generator when voltage was dropping close to 12.2 12.0 zone. Especially if the furnace was to run overnight and we are not leaving in the morning. LED lighting was a huge improvement on voltage and effective battery capacity. Verify your converter is charging at 14.2+ volts (at fuse panel) to minimize run time.