To answer the OP's question, yes, there is enough room to access the spare tire winder with your trailer hooked up. I had a flat on the LR of my F150 last fall while towing my rig to my "Snowbird" spot in TX. You use your truck's door/ignition key to unlock the access hole (located under the flap above your rear license plate). You assemble three sections of the jack crank, insert it through the hole above the license plate and guide it into the receiver above the spare tire. Then crank down the spare.
Be careful, though, and have a spare key handy. My key is still stuck in the lock plug. Obviously, I didn't replace the plug in the hole with my key stuck in it. Oh, and take a picture (with your phone) of the jack box with the jack in it BEFORE you remove the jack. Use the picture as a guide when you put the jack back in the box. IT'S A BEAR! and the box won't close if everything isn't in its correct location!
Best of luck, and hope you never have to use your TV spare while towing.