I had the batteries tested a couple times in the past. The last time I took them in, Interstate found a bad cell. This was the first trip with the new unit and I was really hopeful I'd have it all sorted. The battery started at 12v when we left the house. After 4 hours of driving, we overnighted and were at around 11.8-12v (so no change). In the morning we were down to 11.5-11.8v (ran the furnace). After 10 hours of driving we were still at 11.5v.
I stopped at Napa and bought a wire brush, contact cleaner, and dielectric grease thinking it must be the cable. Jiggling things seemed to help, but I couldn't be sure...
So, we impulse bought a generator to make sure we had power for the trip. When running the generator we had AC and the charge light above the stove lit, but the battery never increased more than .1-.2v after hours of running.
If the batteries are bad, the dealer will swap them again, but I'm sure the issues has to be something else (converter, wiring between battery and converter, shutoff switch, etc). If it were the truck, then the AC lead would also be suspect since the generator was problematic too.