Your going to make it much harder than you need it to be.
First off, I keep seeing a reference to "Screen material" but no actual description.
Secondly, the 5/8" to 3/4" is ambiguous. If you used Poplar, it was undoubtedly 3/4" thick.
Thirdly, it's not advisable to try to make a Frame first, them apply the screen to the Frame.
For one thing, you cannot see the "business" side of the screen when it's upside down.
Another issue is that the Frame would have to have Corner "L" Brackets, which themselves will create at least a 1/8" minimum stand-off...or more if you cannot get the screws for them to set flush with the surface of the Brackets.
Then to protect the screen surface you'd have to have something soft and clean underneath it.
And lastly, attaching the Trim correctly from the Rear is just a wrong way to go about it. Besides the difficulty of attaching the Trim securely by shooting fasteners through the Back of any supportive material and into the thinner Trim, if the trim ever had to be removed, the screen would again have to be taken down and laid flat on it's face.
There are several ways to go about this, but really only a couple of correct choices.
Considering that you already have a set 2.4:1 perimeter of 124" x 52", I think it best that every effort be made to retain those dimensions. That being said, 1" x 3" Poplar edging installed at each corner and along the sides, spaced at reasonable intervals will serve to equalize the "surround" to the Screen's surface.
The 1" x 3" x 5" "blocks" are attached directly to the wall with just one screw though the center. You don't have to hit a Stud....just use Coarse Threaded Drywall screws and hand tighten them down to just snug. Best to always pre-drill a hole to prevent splitting, and drilling a slightly larger, shallow hole to allow a recessing of the screw heads is always best.
Now all you have to do is piece the Velvet wrapped Trim onto the Blocks by attaching them using 1-1/2" Finish nails. The Trim's beveled edge will contact the Screen's edge all around, and probably do so completely flush to the surface.