Had the same truck you do but
it was a 2010 F150. Didn't have good luck with it so it's gone now for the newer 2012.
with the 2010, I had 1430 payload along with your specs but the tow rating is 9600 lbs for a 145" 5.5' bed 4WD Crew Cab 5.4L with 3.55 axle.
The 3.73 axle is only available in the "MAX TOW PACKAGE" which is 11,300 lbs tow rating. You and I had the "STANDARD TOW PACKAGE" which is 9600 lbs with the 3.55 axle.
Since you only have 2 people and a dog, you should be fine. Especially if you don't load anything else in the truck.
We have of family of 4 and with everything in the truck including tonneau cover, bed-liner, and all gear in bed is 800 lbs.
The 2010 had 1430 lbs of payload so we had 630 lbs left. Using the average tongue weight percentage of a dual axle travel trailer (13%) and the average percentage of weight transferred from the trailer tongue weight to the truck axles, using a weight distribution hitch (80%), you could possibly tow a trailer weighing 6000 lbs loaded. We just didn't fill the fresh water tank all the way and kept the loading in the 26BH to a minimum. The new 2012 has 1582 lbs of payload so we can load quite a bit more.
2010 F150
towing 19BH and
towing 26BH.
Good luck.