I live in the Smokies and I can not suggest any private campgrounds because I have never camped at one. I have several suggestions, but I also have a few questions which if you answer for yourself it will give you a better idea on where you want to stay.
The Smokies cover 2 states, NC ( The better state of the two,
) and TN. Are you wanting to camp in the woods (boondocking it) and be in the woods and so forth or are you coming the Gatlinburg and just want to be near the strip? These are two different goals completely.
If you are wanting to be in the woods, any of the National Park Campgrounds that accept trailers would work...Cosby and Cataloochee are a bit more remote, but they are peaceful. Cataloochee is in the northeast section of the park and the valley is very pretty. The Park Service reintroduced elk to the Smokies a few years ago and Cataloochee has a nice herd of them now which you can see in the morning and evening hours grazing in the fields.
Big Creek and Abrams Creek are very remote and tent only, but Big Creek has the best darn swimming hole around at Midnight Hole Falls..absolutely gorgeous there.
The NC side of the park you have Smokemont and Deep Creek which offer creeks that are frequented by tubers, especially Deep Creek
At Balsam Mountain the park's highest campground, you are camping at nearly 6,000 feet! In the summer it is cool, no need for AC there at all and is access off of the Blue Ridge Parkway.
Now if you are going to the Smokies to be in the mountains but really dont need to be near Gatlinburg, I must say there are two really nice gem of campgrounds in the area, Davidson River is in the Pisgah National Forest outside of Brevard and I think US Forest Service campgrounds are boondocking in style as they offer hot showers. Davidson River is a great tubing river and just peaceful there. Then down near Franklin is Standing Indian. It is located along the headwaters of the Nantahala River. Standing Indian is in the Nantahala National Forest. It too is gorgeous, not sure any GSMNP campground can compare to it, even Cades Cove.
On the TN side of the park, Cades Cove is very nice, the cove is a great place to explore and better bring your mountain bikes to ride the loop road there. Elkmont is also very nice and the Little River is a nice tubing river for the kids too.
Private, I have seen many of them, but the one closest to the park near Townsend seems nice from the road. I dont know the name of it, sorry.
We grew up camping in national parks in Jayco pup and summer camping in the Smokies is nice even without AC and power and so forth. I highly recommend it. If you are in need of showers many of the private campgrounds around the park will allow you to pay a small fee to shower.
Gatlinburg is nice, but very crowded and Pigeon Forge, well that is another story all together there.