I’m big into projection and have gone up the food chain over time. DIY with paint is good but sounds like not an option due to the wall not being finished. Lots of very cheap options on Amazon but often come with creases out of packaging. Even a sheet will work.
Just the fabric for DIY, Carl’s is very good.
carlofetWe have several DIY projection screen materials (projector screen fabric, movie screen material, projection screen cloth) to suit your home theater, backyard theater and large venue needs.
https://www.carlofet.com/shop/projector-screen-material[https://www.carlofet.com/shop/projector-screen-material?screen_material=FlexiWhite&aspect_ratio=16%3A9&finished_edge=Raw+Material&seam=Seamless&packaging=Rolled+on+a+Tube&size=145%22+%2871x126%29]
Fabric + screen frame, if you want something nicer (over your price range, but good enough to compete with most options less than a thousand, I used to have this one).
amazonBuy STR-169120 Silver Ticket 120" Diagonal 16:9 4K Ultra HD Ready HDTV (6 Piece Fixed Frame) Projector Screen White Material: Projection Screens - Amazon.com ✓ FREE DELIVERY possible on eligible purchases
https://www.amazon.com/dp/B00CYLOTPK/ref=nosim?tag=cougarboard07-20[https://www.amazon.com/dp/B00CYLOTPK/ref=sspa_dk_detail_0?pd_rd_i=B00DGVHZSA&pd_rd_w=xyIM1&pf_rd_p=7d37a48b-2b1a-4373-8c1a-bdcc5da66be9&pd_rd_wg=a3QlL&pf_rd_r=AN36FCKWXVBD34QF6XV2&pd_rd_r=cb5be47f-d77e-4508-97a5-85c1682213f6&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEzMzNHTFFBMlUzWFNHJmVuY3J5cHRlZElkPUEwMTk3ODU2MlVBREgyVzNOQUhXQSZlbmNyeXB0ZWRBZElkPUEwMjgwNjk5Vkw5SkYyRTFPR0kmd2lkZ2V0TmFtZT1zcF9kZXRhaWwmYWN0aW9uPWNsaWNrUmVkaXJlY3QmZG9Ob3RMb2dDbGljaz10cnVl&th=1]