I was just up there for New Years and on the kangamangus hwy that cuts through the national white mountain forest. I would be a little apprehensive about taking a trailer through that cut. It's doable, but there are some hairpin turns, climbs and descends that one might find tough. There are no gas stations for 30 or so miles once you pass Loon on the Lincoln side until you hit the end in Conway. That being said, in the national park there are several CG's along the way and breathtaking views. But I think they would be more primitive being in the National park itself.
If you stick to the 93, Franconia notch side you have the man in the mountain and Cannon tram which takes you up to the mountains during all times of the year for site seeing. If you go to the Conway and North Conway side your more setup for Mt Washington. Both sides have their attractions, so you might consider a 2 stop trip over a longer week, like 10 days which would cut your driving time down if you want to see a lot of the area.
Sorry I cannot recommend a specific CG, I thought possibly adding attractions might help in your decision. The 93 side offers a little easier access and less State Road time, but might also be more crowded as a result.