Good luck and be safe. If you're not aware, there are several weight limits for a tow vehicle; gvw, gawr, gcwr, payload etc. that will probably come into play before you hit your "tow rating"....which has become more or less a sales gimmick anymore. Look at the numbers on the sticker inside the door, particularly payload. You mention that you will probably be towing the trailer "with hardly nothing", but if you added batteries and propane you added a significant amount of weight; then, kitchenware, pots/pans, coffeemaker, clothes etc.; they all add up quickly and you will probably have hundreds of pounds added....and the longer you have it the more you will accumulate. You will never know actuals unless you hit a scale (very advisable).
You mention you have weight distribution bars "as well as a sway bar". You don't mention what kind it is but you need a VERY good one for that setup. If it is a sway bar that attaches on the side using its own little ball....I've never had much success with those, and with a lot more TV and smaller trailer.
Just look those things over I mentioned and have them in mind while you take your trip. Keep that video in mind as well. When they start wagging they can be hard to stop. Hopefully your OTR experience would help you there if, heaven forbid, something like that starts. If the tail of the trailer EVER moves independently of the direction of the TV you have a problem.
Not trying to be a sour note, just wanting to give you all the info possible to keep you and yours safe. In my case, once knowing all the numbers and the knowledge that I did have occasional sway (and slightly over payload), those numbers played in my head every mile I drove along with visions of what could happen every time the trailer twitched. It drove me crazy. When I got home from our first cross country trip I went straight down and bought a new HD truck. The risk was too high and a new truck wasn't that much in comparison...and the truck I was trading in was virtually new....
Again, I wish you and yours a safe, happy trip; make some memories with those girls - they last a lifetime.