Riding the BRP is not all that hard. As mentioned, there is no fuel stops on the BRP, but are some right at some of the exits. I tow our 31' TT on it with no trouble. It is in fact one of the better roads thru the mountains.
However, It is slow, Real slow. You can do it in 15 or so hours IF you don't stop. But if you don't stop, what is the point? To do the whole BRP, and see even a little of the sights. You will need a week.
As for CGs along the way. There are many National Park CGs with no hookups, but do have dump stations, and nice restrooms. Linville Falls, and Julian Price, are our favorites. There are at least 3 private CGs with BRP access. One near Little Switzerland, (Bear Den). One at Glendale Springs, (Raccoone Holler), And one near HWY 18. (Miller Camp). those 3 have full hookups.