You'll get different opinions on this but I'd vote for a quick answer of 12". It looks like you will have a flat simple floor which makes life easier, the pop up roof might be payback but is a cool idea.
You will be a foot wider than a 6x12 box cargo trailer but the ground clearance and frame structure could be somewhat similar. Cargo trailer manufacturers usually go with deeper frame sections when they build 7' wide but you won't have that need, that's why I'm making the comparison.
One of the things that you have to plan out is your running gear. With a trailer that size I'd go with 15" wheels and be done with it, yes 14" will also carry the weight just fine. I have a 6x12 cargo trailer and it has 2"x3" frame rails, 4" drop 3500# axle located over the springs with fairly short spring hangers....12" under the frame. That's about as low as they could get it with standard hardware...and the axle travel is only about 1-3/4" before if hits the frame, which I'm not totally wild about, but I guess it works. I looked at replacing the spring axle with a torsion axle but to keep the same ride height you have to use an axle with a 10 degree "up" starting angle and that also wouldn't allow a whole lot of axle motion. Mind you it will work, it's just far from ideal. So some of this comes down to how you rig your suspension.
My brother has a 6x12 cargo trailer that has drop-down supports at the rear corners that have only about 10" of clearance and he doesn't seem to drag them in normal use. I've seen some motorcycle trailers that were really slammed but I'd worry a bit about dragging a corner with one of those.
Hope this gives some food for thought.