Welcome ! Newfiescottr !
Even if you have the stock transmission cooler, it may not be adequate for pulling long hours with a loaded trailer. The max is 5000 lbs with the correct hitch, and many have experienced hot transmission lights, which is not good. You have to pull over then shut off off for a good while, to let it cool down.
Do yourself and the transmission a favor before you do this trip. Get a much larger transmission cooler! I would consult with your dealer to do two things, ask if they could install a new transmission cooler and do a proper transmission flush before you leave.
A happy trail & pulling rig will have fresh fluid and a better than stock cooler.
Alternatively any good transmission shop could install the cooler, but still I would have the dealer then do the flush, as they have the computer's to re-set the TCM to let the ****** know you got new oil and begin to learn the new shifts.
If I was doing this trip I would also find a a bottle of red LubeGard and add this to your transmission fluid after the flush. This is an excellent friction modifier
http://www.lubegard.com/~/C-112/LUBEGARD+Automatic+Transmission+Fluid+Protectantwhen added to the
ATF+4 fluid that the Jeep Liberty uses in it's transmission, this will help the oil stay a bit cooler.
The way to confirm that you have the cooler is to trace the transmission cooler lines forward to the very small (rad) cooler, or back from it.
In the long run it is cheaper to install the new cooler and do a flush, as compared to having to rebuild your ****** because it fried from your heavy load and the heat!!
Three other recommendations are change your rear differential fluids to the recommended synthetic GL5- 75W-140 gear oil for trailering, and take the overdrive off to stop the roaming when climbing hills and use a one grade up higher octane fuel when pulling.
Best of luck on your trailering and your cross country move! Your KJ can do it with the above completed and stay under the 5000 lbs.
Note if your trailer has trailer brakes, (it better) you will need to ensure that they are properly connected to the KJ.