I don’t know about renting a projector or a 10k lumen projectors, but you definitly want through as many lumens as you can afford. If you can cover the screen with a dark tent or portable stage on a trailer, you would be able to stop direct sunlight on the screen and increase the picture quality. You also may want to use a projector with a rear projection screen.
I use Boxlight Projectors for high lumen applications. Their Pro4500 is 5000 lumen, dual lamp projectior that retails for $6k. The MP-65e is a good value with 4500 lumen at $4k.
Good luck....
Shannon