Freddy...you made use of existing material. Granted up close it does have a "rough" look, but in the dark...in my opinion not many will notice.
The idea of a carrying bag is a good one. However, you may need a heavy-duty sewing machine. But, if you were to go to a canvas shop or seat-covering shop with the materials cut to the size needed, I bet they would sew the pieces together for you, in a fraction of the time.
Some of the moving blankets are thick and some are just the right thickness. I believe Northern Tool + Equipment has a "thin" moving pad. Not sure about Lowe's or Home Depot.
You might consider making it like a laptop bag that has two inside inserts, if you only have a tri-fold facade. However, if you have 4 pieces, then I'd go with two bags; each with only one insert. Four pieces would be a bit heave to lift and move around, whereas a 2-pc bag would seem about right. I would have the back made so that you could fold it over the top to the front of the bag and secured with Velcro straps...or use ratchet straps.
I like the concept. The light behind the spandex does look pretty bright. What are you using for the light?
Regardless, it's functional and will do until you decide upon modifications.