You definitely need the brakes working. You're not legal without them.
I have read somewhere about some trucks having a problem with some trailers. Found this online: "Ford vehicles, particularly certain F-150 and Super Duty models, have experienced issues with their factory-installed trailer brake controllers. These problems can range from the trailer brakes not engaging at all, to intermittent connection problems, or even the trailer brakes applying unexpectedly and forcefully. A common cause is a software glitch within the integrated trailer brake control module, which can prevent the controller from correctly detecting a trailer or properly applying the brakes. " See if you can borrow someone else's truck to see if the problem persists.
Another likely failure point is the 7 way plug on the trailer. Those contacts can become spread out and/or corroded. Look inside the plug and inspect the brake contacts with a flashlight. If they're corroded, a temporary fix can be using a thin file, emory board, or screwdriver blade to scrape the contacts clean. If they seem spread out, use a small screw driver or hook to bend them back inward a bit. These would be temporary fixes.
If your plug is shot, it's easy to replace. Disconnect the trailer batteries before you start because the 12v Power lead is hot and then follow this wiring diagram to install a new plug.