You have a Tahoe, drive either in tow haul mode or with the HEADLAMPS turned On. That will cause the PCM to keep the generator charging at a higher level than if tow haul is not selected or headlamps are off (daytime anyway). Normally, the PCM sees the TV's battery charged after a while, notes the needed load and drops output down considerably. With tow haul selected or Headlights On, PCM keeps the boost up. (GM, can't speak for others.)
And yes, check to make sure the wire is connected and if a fuse is installed. I ran into that problem with my 2013, the wires were disconnected from the factory, dealer didn't connect (strange, the fuse was there), I didn't check, I ran the DC fridge from UT to Glacier and guess what....both batteries were dead. it was hot, I was hotter (and a bit peed off; ok REALLY peed off), so I just got out the ratchet and drill to raised the roof, and when it was allowed, connected the generator and charged things back up. Next day, I spent some time in a small town's auto parts store, getting nuts to connect the wires to the correct posts.
As for the battery...yes, should be NO problem. In my driveway, rather than spend five or more minutes to connect my batteries, I sometimes use a jump box to raise and lower the roof. One day, I probably raised and lowered the roof a half dozen times when I was adjusting the winch limit switches and the motor never appeared to slow.
The jump box is a dedicated jump box; now one of those that have a built in air compressor/light and from past experience, it allowed me to crank over an H1 diesel engine for about a hour when troubleshooting a starting problem. (I now travel with the jump box; I have never used it on a trip, but have helped out others with similar problems.)
And you should not get stranded. It was pointed out to me, that the newer power winches have a manual bolt to allow lower/raise without power. Just make sure you have a power drill with the correct socket (and extension if needed) for those just in case things that do happen with power winches.