Just reporting back... We ended up bypassing Little River Road after all.
We drove down Little River Road without the trailer to visit the park (we were staying in Townsend at Mountaineer campground). We decided not to go that way with the camper. The road is narrow, there is no shoulder and there is a significant drop-off from the pavement in several places. Also, several curves were pretty tight with oncoming traffic on one side and a wall of rocks almost up to the pavement on the other (read that: no room for error). I didn't think my driving skills and my nerves were quite up to the task
I'm sure others would be fine.
Instead, we routed up through Pidgeon Forge (which was interesting in a more-Branson-than-Branson sort of way) and then took the Gatlinburg Bypass (which was beautiful). Hwy 441 through the park was fine, with frequent pullouts.