So if you can’t find something to screen as well as you would like, have you considered a shenel (I could be spelling it wrong) and a kula cloth? Very popular with the backpacking community.
And like Silje I just find a bush and go. Trails, parks, golf courses, etc., just find some coverage and be quick about it. And watch the splatter and try to angle it away from your feet (we used to camp and hike a lot when I was a child).
I personally didn’t do this but my hash house kennel was running around town and one lady really had to go. So she jumped behind the shrubbery at a Rite aid while the rest of us stood guard increasing the privacy. This was after we had went inside and sang to the pharmacist who was a friend of one of the hashers. 