IMHO, I'd just go buy a cheap 40-50 inch LED television and toss a cover over it when not in use.

Something like:

https://www.amazon.com/Outdoor-Cover-Weatherproof-Dust-proof-Microfiber/dp/B0744MHRMX/ref=sr_1_1_sspa?dchild=1&keywords=outdoor+tv+cabinet+weatherproof&qid=1615573870&sr=8-1-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEyMFg2Qjg4RFJNMEIxJmVuY3J5cHRlZElkPUEwOTY2ODI5MjgyRlBCOEVNWlNIQyZlbmNyeXB0ZWRBZElkPUEwMjg2MDgzMlFNRkZFQUtCQ1k3SCZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU=

Projectors would only look decent in low light/shade and you can get a TV of the above size for not much more. and it's going to look much better in the daylight.