Tow capacity and haul capacity do affect each other and this is a zero sum calculation! Your Jeep and trailer (you didnt say how you are towing the Jeep) will reduce the amount you can haul in the bed. Likewise, the weight of your Palomino Backpack in the bed will reduce the amount you can tow.
You really are better off with an F250. As mentioned, 1200 may be the dry weight but when camp gear, clothes, water, food and fuel are added, you really are at or near haul capacity. That's not the best way to be driving around the country. Nevermind the fact you are also towing the Jeep.
Could you do it? Yes.
Will you be white knuckled and nail biting on hills, curvy roads and mountain passes? Yes
Would your truck last, doing that regularly? No
Your brakes would warp, your stopping distances increased, your frame and springs would be near maxed out all the time, sagging...your transmission would need servicing under the Severe schedule.