I agree that an F250 or other 3/4 ton diesel is the best choice for pulling that size trailer. However, Denna's question was more focused on brakes than on getting up the grade, and there too the larger wheels & brakes on the 3/4 ion diesel will make a big difference. And they can add an exhaust brake as well, to give the service brakes some relief.
That 9300 lb towing capacity on the 150 assumes the truck has nothing in it but a full fuel tank and the driver - no optional equipment, no passengers (are you staying home?), no gear in the bed, and not even a trailer hitch installed. The actual capacity of your 150 when loaded for the road will be substantially less than 9300 lbs. And it assumes that the trailer will have great brakes of its own, since the truck is designed to stop only it's own GVWR, not the weight of the trailer too. Most trailers come from the factory with skimpy brakes, barely adequate for the weight when new and deteriorating after a year or two.
That trailer also has a 690 lb tongue weight when empty and probably 900-1100 lbs when loaded for a trip. That's a lot of weight for a 150 to carry, even with a weight distributing hitch. Odds are you are going to exceed the truck's rear axle capacity (GAWR) and maybe the GVWR as well, once passengers and gear are added to the tongue weight.
Denna, you call tell hubby for us that he is being terribly optimistic about the F150's capability and also that he is taking chances with his family's lives. Bite the bullet and get a F250 diesel - it is designed for those kinds of loads. And I suggest adding an exhaust brake as well. Besides, what man doesn't want to have a bigger truck?