I've done grilled pizzas a number of times. I made a bunch of dough balls to take camping last year, and made some white pies with fresh tomato, and some margherita type pies with san marzano tomatoes straight from the can. I use a portable grill with a cast iron grate. The spacing is small enough that I can brush on some olive oil and drop the dough right on the grate. You could use a screen, though if the spacing is really wide. I usually brown one side, then remove and flip it over, top it, and put it back on. Works very well. I made smaller dough balls, and they lasted all week in my cooler.
I use some semolina in my dough, which makes it less sticky. I can put the dough ball in a ziploc bag without any flour or oil. I just rip the bag along the seam and it separates easily from the dough. I had everything ready to go in my cooler, and it was so fast and easy to throw a pizza on the grill whenever we were hungry.