There is some pretty good advice above. What I can add is,
Fully understand the GCVWR of the Jeep and the true weights of the loaded camper. I couldn't be sure of the tow rating for your TV. Edmunds shows 3500 lbs for the Unlimited with 1000lbs in payload.
My 122S weighs 2800 lbs, filled with water and loaded for one to camp. The tongue weight is 480 Lbs, so it is very hard on the TV rear end. Your's will probably weight more since it is heavier to start with and have gear for four.
That tongue weight gets subtracted from TV payload when loading the gear inside. Thankfully the A122S has a good amount of storage space while on the road. Tongue weight adds more than its number to the rear axle, taking the addition from the front. Most jeeps I've been in have a softer, long travel, off road suspensions, you will probably need a weight distributing hitch to have a safe ride.
Consider the tow capacity vs the road terrain where you will normally be camping and where you would like to go. You camp for fun, and it is no fun driving a overburdened vehicle that is difficult to get up to highway speed going up an onramp, on a regular basis.
If you are comfortable with this, then go for it.
An A122S will not comfortably sleep 4 adults. The front dinette sleeps two relatively young children. So if you're family is OK with that, it won't be a problem.
2012 Rockwood A122S
2008 Toyota Tundra Double Cab 4X4
Former owner of a 2002 Coleman Niagara GTE