Smallest TT with bath and room for 2 kids?

A Casita sounds like it would just barely house you and the kids, and would pull great behind the JK. bgum mentioned the fiberglass egg trailers that Casita is one of. A nicely optioned Casita 17 foot spirit deluxe might be a great start.

Excellent choice would be a Lance 1575

Link To 1575

Have you considered a popup trailer? Might be the best choice weight-wise.

Jayco has a new Hummingbird coming out - the website doesn't show the weights yet, but it's smaller than the current ones and may be an option (it's a hybrid).

If you're really set on a hard-sided, you might look at skipping the wet bath and using a cassette toilet and a separate camp shower.

I would rent a smallish TT before you buy one. You may be, barely, under your load limits, but would NOT be happy with the towing results.
I have towed with a Dakota pickup truck that had a 3.7L engine. I was well within it's stated limits, but that poor motor just screamed at me the entire trip. I would not do that ever again. The next week I had a 3/4 ton MegaCab diesel!!!
I'm not suggesting that you need to do that, but just be forewarned that the JKU will struggle. You need to experience what I did to understand.

there is a KZ 14 RBT - very nice!

we have the 16 RBT which is maybe even more than you need...

JIM in NH

Anything with the label something close to "16RB" comes to mind.

Have a look at these ones:Starcraft 3500 pound gross weight trailers

Hard sided would be 16ft Airstream Bambi 16. Sleeps 4 with wet bath. Right at 3500# gross limit? Try some of the fiberglass egg style trailers.