Adding a trans cooler will be cheap insurance to protect your transmission, but it will not make it tow any easier.
I added a TransGo shift kit and upgraded separator plate to my Trailblazer's transmission. I have not towed since I did these upgrades but shifting and the torque convertor lock are dramatically improved. If your not comfortable with doing your own work, this might be a bit overwhelming.
I don't know what is available for your F150, but many speed shops offer PCM reprogramming to improve performance. Often referred to as a "tune", they can tweak it for towing, better mileage, whatever you like. Keep in mind that this also voids your warranty if you still have one.
As someone who is towing a trailer that is at about 97% of all my rated capacities I can offer a bit of advice on towing. If you have "P" rated tires on the truck, run them at least 5 psi over what the door sticker says. Experiment a bit, 10 might be better for you but don't over inflate them. If you can, get LT rated tires instead. Much better for towing. Shift as much cargo as possible from the TV to the trailer. Everything that eats your cargo capacity diminishes the truck's ability to tow. Believe it or not, moving that weight to the trailer is easier on the truck. Tow in 3rd gear (or what ever is not over drive). The keeps the trans from shifting a lot and reduces heat. Keep your speed modest. I used to travel at 62/63 at 2700 rpms with our old 3500# trailer. Ran beautiful, got 12 mpg, and towed smoothly. That same speed and RPM with our 4750# trailer gets me 8 mpg. I now tow at 60 mph and 2500 RPM. Tows beautifully, I get about 11.5 MPG, and it hardly shifts even on hills. You'll have to experiment to find your own "sweet spot".
I'm jealous about the Trundra. Those are nice. So are new F150s!
2013 Jayco Jayfeather X20 E, 2008 GMC Sierra 1500 Crew Cab Z71 SLE