Could you fold your flat into the preferred fold and then sew just one line to prevent it from coming undone (and not have to keep folding it)?? I was hoping it would hold it in place but since only one seam would be done it would allow it to still dry quickly.
I'm pretty sure you can fold, sew, make the diaper any number of ways. I sewed about 40 diapers for my daughter out of old receiving blankets, towels & clothes & stuff...my husband fondly called them the "crap rags".
I always thought about doing that when my youngest was in diapers. I thought I could just sew up the middle of an origami fold and I wouldn't have to fold it every time. But then I figured it wouldn't dry as quickly, and since that was the reason I used flats to begin with, I decided against it. I also enjoyed folding it, but that is just me.
Give it a shot with one diaper, try it for a few days. It's as simple as ripping out the seam if you don't like it. Let us know how it goes. You also might get better responses in the diaper making forum,