The answer to the question of keeping or bypassing the stock trans cooler in the radiator depends on the ambient temps you'll be using the truck in. The stock cooler also doubles as a heater for the trans to get the oil up to temp quickly on cold days. Colder is not always better when it comes to oil, the trans is actually designed to have the atf at about the same temp as the coolant most of the time. That being said, if you drive the truck primarily in hot ambient temps and/or it isn't a daily driver, just bypass the stock cooler. Make sure you use an aftermarket cooler with a sufficient tube size, otherwise you'll cause pressure issues in the trans. A stock radiator for the towing package will be fine, just make sure the rest of the cooling system is up to the task. My truck never has issues running hot while towing a 5000lb car trailer, including extended up hill pulls at 60mph (about 3/4 throttle in 3rd, mine is only a 5.2) with the A/C on in 90+ degree heat. The coolant maxed out at about 215, which is perfectly acceptable.
You may have already addressed this, but the part that makes me cringe the most reading that you plan to tow so much with a 1500 on steep, curvy roads is the brakes. The stock brakes are not up to that task. Is the trailer 7000lbs or are you figuring the combined GVW will be 7000lbs? If the trailer itself is 7000lbs, I think you need to reconsider your choice of tow vehicle. My truck, with the stock brakes, would struggle to stop that 5000lb car trailer quickly at highway speeds, and it had (surge) trailer brakes. I nearly caused a big accident on the highway once when traffic stopped suddenly and, despite leaving several car lengths in front of me, I barely got the truck stopped in time. Standing in the brake pedal didn't even generate a chirp from the tires... The stock brakes didn't have enough power to even try to lock up the tires with that much weight. I upgraded to larger pistons on the front calipers and rear wheel cylinders (all bolt on stuff) and used the most aggressive "street" pad compound I could find (Hawk severe duty pads) and that helped significantly, but I still wouldn't trust this truck with these brakes towing a 7000lb trailer, let alone on a curvy, hilly road. I've you're figuring 7000lbs total combined weight, you'll be fine as long as the brakes are good. The trans and engine will handle it just fine.