Your thought process is solid.
Idk about that particular extension, but with sufficient diagonal chains, I wouldn't have issue with flat towing a Jeep with your truck/camper combo.
Make sure you get a 2.5" square extension. Don't use the hitch sleeve adapter. Those are problematic with heavy or long term use.
No aux brakes needed unless you really want them, and at that, if you felt the need for a surge system on the toad, you may consider a compression brake for your truck vs the aux braking in the jeep. Pac Brake load leash.