painting it before it is stretched is an option, but that can get a bit wonky when the carps go to stretch it. I would still try to keep in on the floor if you can, but I would also do a test piece to get your technique down before you throw the entire thing up.
Scrim is a very different thing to paint. We had a large paint frame at the last place I worked and the painters still painted all of our scrims on the deck.