That is pretty close to what I was pulling and had no problems. Saw alot of people with larger rigs than that. We just got back, and there was a bit of road construction in Cherokee that slowed us down a bit. We love the Smokemont campground, but have stayed up in Pisgah. Smokemont is nice, site 34 in F Loop is by far the best as it is right on the river. Sites 34 - 44, even numbers only, are the only real river sites. What I love about 34 is the small steps down to the river someone made with stones.
Shouldn't have any trouble. People take much bigger rigs to the area.May be a little slow going up some of the hills, but other than that have fun. If you need more info, just let me know.
Late spring before schools are out is a super time to go and camp in 1 of the national park campgrounds(no hook-ups). April could be cold at times because of some altitude so May might be a little safer but I would risk later Apr also.