Good advice in the replies here regarding battery usage and staying warm.
Are you planning on staying in the developed campgrounds (no hookups and what I would term 'dry camping') or truly boondocking in a dispersed site off a USFS road? To the best of my knowledge, only the North Mills River and Davidson River Campgrounds are open year round; the others close on or before October 31.
If you're thinking of dispersed sites, running the genny whenever its needed shouldn't be a problem (assuming you don't have neighbors) so even at ten hours per day for five days, you're under $20 total - take along a small electric heater and you should be good. You didn't mention what type RV you're renting or how familiar you are with the Pisgah area but most of the FS roads are not what I'd call 'RV friendly'. Also, five days may stretch your limits of your holding tanks unless you find somewhere to dump/refill and freezing temps could be an issue with these as well.
With this being your first trip, my suggestion would be to stay in FS or NPS campgrounds for your dry camping experience. You'll have access to water and toilet facilities and, this time of year, you'll likely be able to find a site far enough away from other campers to get a 'wilderness' experience. In addition to the two Pisgah facilities I mentioned above, Smokemont and Cades Cove are two NPS campgrounds in that area that I'd recommend.
The NPS Cataloochee campground isn't open but well worth a day trip. Go in late afternoon and there's a good chance you'll see elk and hear the bulls bugling.
Have fun and please post back with pictures and a trip report.