Grace, resist over-thinking it. By all of your accounts you have a capable truck with the right equipment. If you have tow mode, you have the factory tow package that includes a heavy duty trans cooler. Adding airbags will help keep the rear end from sagging but they do not increase tow capacity, so consider those optional. The biggest help you can do is changing to 17" wheels and load rated tires. That should get you 1000 lbs roughly by itself. The next best thing you can do is to make sure your hitchwork is installed and setup properly. There are procedures for that because it varies for each specific truck and trailer combo.

All that said, none of it will help you climb a mountain with ease and increased speed. All of my stuff is installed right and i still find that the truck is most comfortable at a top speed of around 60mph towing on flat pavement. I plan my travel to stay off interstates for this reason and i find i get much better mileage as a result and it really doesnt take that much longer to get where i am going.

 

Every one has a transmission cooler if not we would be broke.
Its how big is the one yall have. Its right in front of the Radiator cant miss it.
And for the Super Filter aka CAI, it wont do anything for towing, but if you buy a K&N Drop in Air filter it would brethe a little easier but not worth the money when you are trying to solve other problems