So are you asking about minimum framing with an inner skin of some type and foam insulated walls (a composite structural panel) or just some studs, the plywood outer skin and the canvas?
If it is the later, my experience with 5 mm ply suggests that you will need more support to keep the panels flat, especially in the roof where it will tend to sag.
On the other hand, with a minimalist approach, you will gain a lot a leeway on weight and can afford a few more sticks. I would plan on at least 1x2's on edge around the door. For the wall studs that are just stiffening the wall panels you could probably get away with 1x1 (3/4 square), but I would go back to 1x2's where there will be shelves or cabinets, and at the bulkhead/galley shelf. I like to use #6x3/4 FH screws, and anything smaller than 3/4 inch wide gets a little sketchy, IMO. If you have to butt join any panels where they land on a spar or stud I would back the joint up with a strip of the same ply, and shave the stud down to back that up (rather than going to a wider stud for the ply to land on), similar to how I did one of my ceiling panel/roof spar joints. From my experience, if you try to butt a panel on the edge of a 1x you have to staple too close to the edge of the ply and it can blow out the fibers along the edge leaving divots.
Something else to consider is how you will attach the walls to the floor and the joint between the walls and roof. The canvas will wrap the corner of the profile much easier with a generous radius, so you might want to plan for a full perimeter cleat along the profile.
For the roof, depending on how tight your profile curves are, you will probably need 1x2 spars on edge every 7 or 8 inches (at least in the flatter areas) just to keep the ply from sagging under it's own weight; it won't follow the profile in the middle of the span without adequate support. As a point of reference my roof is 64 inches wide and has a 520 inch profile radius at the ceiling. At first I thought I could get away with just two 2x2's at the vent, using the cabinet face frames at the front and back, but I was wrong. Then as a test I laid the ceiling over with additional spars at 12 inches, but it still sagged. Finally I settled on every 7 to 8 inches and it is holding shape well.
Another way to save weight is to pick a lighter species of wood. I used a lot of clear select cedar, which is very light and rot resistant, and some poplar for my framing, only resorting to heavier woods like oak for the hatch hinge spars and Red Grandis for the hatch ribs (... and a bunch of soft maple for cabinet frames

... not light but looks really good

)... but fir is a lot less spendy. The trade off is that you can almost always use all of the better wood. Whereas if you try to rip your structural stuff from cheap box store lumber you will end up loosing a lot to knots and other defects.