Well Steve, the E rated tires will help with the heavier trailer. You may have 1,000 lbs of tongue weight, when loaded to camp with that new trailer. I think the truck will pull it without issue, but you will have to watch your payload. With the tongue weight, you may only have 500 lbs left for everyone/everything else in the truck.
What does the yellow/white tire loading sticker on the door show for payload?
I am sure there are others towing a similar size trailer with the Eco-boost 150. The newer F150s are great tow vehicles, but with the right tow package, some are better than others.
Jerry