Hi Brandon241,
Texas to Mt. Rushmore can take several days of driving for most people. If you're plugged in to shore power overnight, and your tow vehicle's umbilical connector is supplying 12V while towing, your batteries should be fully charged most of the time.
If you're staying overnight without shore power, you'll draw the batteries down some, but they should get some charge back while towing. You probably don't want to try 3 nights in a row without shore power as the batteries may not fully recover while towing.
If using the furnace, you'll draw the batteries down much faster, but should still be good for an overnight stop as long as you get fully recharged the next day.
You can read more about how the residential refrigerator is powered in our owner-written
Residential Refrigerator Guide.