Projectors are a VERY poor choice for any outdoor use, even shaded. You will be fighting the environment very much. It would be worth you borrowing a projector, if at all possible, and trying it out before you set everything up. I'm not sure what the ceiling setup is, but wiring will need to be put in place for speakers and video to the projector as well as power which will need to be considered. But, I would first borrow something if you can or buy a cheap (used) business class projector that's fairly bright and see if you have enough light from it to actually light up a screen at a level which you would actually enjoy during the day.
Keep in mind, you've never been to a movie theater in your life which features 'nice large exposure to the outdoors', except for after-dark drive in theaters.
Outdoor, daytime viewing is impossible. But, if you are able to drop the light output enough in the covered area, you can get a usable image. But, your target contrast ratio is going to be about 12:1. That's all you can realistically expect with a 120" screen in such a space.
I would lean towards the Epson 3800 which is bright, but maintains decent contrast after dark. It's a decent all-around model.
For a screen, a basic Silver Ticket matte white screen will be acceptable. ALR screens won't do a lot in such an environment and will cost a fair bit.
I would more likely recommend a large screen television as even the cheap ones will have far better brightness and image quality than a projector will in such a space.