I did one cross country trip, from OR to Northern NY.
Before the move, we towed a lot in the coastal range. Truck and trailer combo did very well.
My trips now are from NY to Virginia, Pennsylvania and other regional areas within 300 miles or so. The Solitude pulls very well and feels stabile behind my truck. I've hit some high winds in Wyoming and never felt pushed around.
Semis passing by do not push me around either.
The 310GK is probably the most popular Solitude in the line-up.
The general 5th wheel rule is to estimate ~ 20% of weight on the pin. At 15k, estimate 3k pin weight, and that gets you close.
How and what you load can also have a significant impact. Firewood in the bed, items in the front/belly storage, etc. generator, batteries (I went with LiFePo, saves weight and more energy).
I'm loaded pretty heavy (pull-out tray in belly storage, Green mountain pellet grill, cast iron pans, etc.)
I do NOT pull with water in the tank, unless absolutely necessary. I do put a few gallons in there in the event we need to stop for a bio emergency

The fresh water tank is slightly behind the axles, at least on my 344GK. So while that might relieve some pin-weight, I would not recommend it.
While a DRW is the better tow vehicle, the 310GK is within most SRW 1 ton pickups capability. I would be comfortable yanking it around with either my '14 3500 SRW or my '21 3500 SRW.
One consideration for you is the Solitude is a wide-body trailer (101" wide). Standard trailers are 96". This extra 5" does reduce turning capacity on short beds. Additionally, the Solitude pin box is not extended very much, bringing the cap closer to the cab. I was OK in most scenarios, and am always VERY careful about the cab clearance with a fixed hitch. I ran into an issue once turning into a campground off from a 55mph highway, the turn was over 90 degrees and sloped downward. This cause me to "kiss" the cab. Was done and immediately went to the slider. No issues afterwards
One more unrelated note; if Disc brakes are an option, GET THEM. Don't think about it, consider it, ponder it, etc. Check the box and move on