I am envisioning a swing-tongue frame, which actually attaches to the
axle on the OUTSIDE of the wheels. That way, the A-frame can be flipped over the roof, from one end to the other, depending on need. It would, of course, require a swivel coupler. This way, you could PULL into the campsite, putting the door on the desired side. Then, flip the A-frame the other way, hitch up, and PULL away.
I'm not actually an engineer, but I met one, once.