I tow a similar size trailer with a midsize SUV rated for 7,000 lbs. It does it, and has for many years, but I'm definitely over my payload (tongue weight+WDH+people+stuff in vehicle) and my kids are 2&4, so they have lots of growing to do.
For this reason alone I'd say you should look at full size SUV's only (since you've ruled out pickups). Especially when you are towing with your most precious assets in the vehicle with you. If you only tow 10 miles down the road to your local state park, then you can get along with the Jeep GC, but I personally don't like to be limited on distance due to an inadequate tow vehicle.
RoyBell has it right, tow ratings really should be viewed carefully, the #1 item to look at is TV payload capacity, which you'll run short on long before max trailer weight when pulling RV's.
Hope the medical issues are manageable, but glad to see you getting out there with the kids.