Sounds too fun! When is movie night?!
I would check out a used screen and just get a table top projector that puts out a fair amount of light. Then just get a coffee table for your outdoor room and plug an HDMI and extension cord into it when in use. Luckily, there are tons to choose from in your price range.
Vinyl screens are easy to wipe down if dirty.
Here is a good site to get you started with your research. There is a screen distance throw calculator for each specific brand and model you check out.
http://www.projectorcentral.com/best-home-video-projectors.htm
