I used 4/4 poplar for the cross-spars, it's light, clean & straight when you cut it from a plank & it's plenty thick enough. Also, a few "prop" sticks & a sheet of 1/4" plywood will hold up a sag in the headliner until the glue sets. With the canvas applied to our birch headliner first, I needed to be careful of the glue & didn't want any fasteners to show when the ceiling was finished. With the side panels cut with a rabbet joint, it was fairly easy to lay the headliner in place with 2 people. Then, there's a lot of blocking to cut to set things in place for the insulation & outer skin.
Looks like it worked well enough for your build, there's always something...