First, it's Smoky. :) Consider that during the day the higher elevations can be smoky/foggy. :) Also, the temperatures can vary 50 degrees over the next month or so depending on whether cold fronts come in or not.
Considering you will ride, plan out your rides and find a central place to camp after that. You could camp anywhere along I-40 all of the way to Newport and ride into Gatlinburg.
With a bike and depending on how far you like to ride, there are a lot of options. The problem with the Dragon is that it isn't near anywhere else in the Park unless you take a gravel road from inside the park over to it.
There are also many public camping areas in both National Forest and TVA lakes although most don't have hookups.