There are f150s and F150s, properly configured, 7000 lbs is not a problem. Any of the full size Pickups properly configured will tow this comfortably.Well, the BIG toy-haulers have the garage completely separate from the living quarters, but you'd better have a serious tow rig to haul those around! I know someone who tows a 7000-lb-ish toy hauler with a F150, but it's pretty much on the limit of what that truck will tow.
The separate garage at the back of the "big" toyhaulers may look separate, but there is usually a door between them, often not sealed, and the two sections share heating and A/C. Any gas fumes in the back soon make their way to the front. I siphon down the bike and let it cool as much as possible before putting it away. I'll run the generator carb dry and let it cool down.
If I do this, the bed in the back over the bike and equipment is usable. I don't think this is much different than would be required to use a van or a shared space trailer.
An open trailer or dedicated bike trailer, or the back of a pickup won't require this procedure.