I bought a used 2008 Lexus GX470 (4.7L V8) in 2015, and I've been towing steadily with it for work and play. It's holding up well would tow a popup easily. If you find a Toyota 4Runner with the V8 (it was optional some years), that's basically the same vehicle. Really a V6 would handle it, but if you want more power the V8 has it.
On the V6 side, another newer Oddy van would work, or a Pilot. Or a Toyota Highlander or a bunch of other SUVs and minivans. A crew cab pickup would do it, of course.
Take a look under the trailer, back behind the wheel. If the axle has a square backer plate with some holes, that's a plate where a brake kit can be mounted. If not, for maybe $500-$800 you still could replace the axle with one that has brakes. It's not impossible.