Tank capacity will also be an issue, I assume you will be renting a Cruise America C32, if so according to their web site these only have a 40 gallon fresh water tank, a 25 gallon black water tank, and a 22 gallon gray water tank. By comparison my 28 ft class A has 80 fresh, 40 black and 40 gray, so we can somewhat easily go nearly a week with 2 people on board, and only moderate water conservation.
Lets put a 40 gallon fresh water supply into perspective with 6 people on board, the big one here is if the onboard shower is used, if each of 6 people were to take a 1.5 minute shower at 2.5GPM water flow, this alone would half empty your fresh tank, and would almost overflow that little 22 gallon gray tank. (6x2.5x1.5= 22.5 gallons), if we add in toilet flushing say 6 people using a total of 2 gallons to flush twice per day that adds another 12 gallons to the black tank, or roughly half its capacity, and we have not even got to water consumed washing dishes, etc. Even if you ban the use of the shower in the RV, your black tank is still going to fill up fairly quickly, assuming you only allow use of the RV facilities during the night, you may last 2-3 days between trips to the dump station to dump the tanks, and refill the water tanks (most national park campgrounds will have dump stations at the campgrounds, and also have fresh water sill stations a safe distance from the dump station (50 ft or so).
As to battery charging, in cooler weather, a forum search suggests that these Cruise America rentals (at least the 28 ft), come equipped with a single group 29 deep cycle AGM battery. If so expect to need to run your generator for at least a couple of hours each day to recharge it enough to make it through the next night. In my experience most National park campground no generator quiet hours runs from about 10 pm - 8 am, the more restrictive ones will tend to allow generators to run for at least 2 hours in the morning and 2 hours in the evening. My advice here is to fill up the gas tank as close as you can to the campground (the generator will consume about half a gallon of fuel per hour), and run the generator while you are doing daytime meal prep, this should give you enough power to run the furnace the following night, though you will probably want to set the thermostat lower than you would at home. Another limitation here will be the size of the propane tank, cruise america only puts a 12 gallon propane tank on their coaches, which might last 2-3 days in very cold weather, or perhaps a week or more with lows in the low 40's and highs in 70's. (assume at full time running the furnace consumes about 1/3 gallon of propane per hour, with 40 degree weather assume it runs about 15 minutes out of every hour) Feel free to do the math from there, note a 12 gallon propane tank, only holds about 9 gallons of propane.
As mentioned most National Park campgrounds have shower and toilet facilities, often this means restrooms in small buildings possibly even pit toilets through out the campground, and a large centralized bath house with showers, toilets, and often laundry facilities at the entrances. Most national park campgrounds are large with hundreds of RV sites, so have large shower areas, etc often near the check in station, which may be half a mile or more from your camp site..
