I have a random Amazon cheapie that I run with a Roku stick. It’s fine for a smallish bedroom and takes up very little space. I filtered my search results for “used” and got an $80 item with damaged packaging for $25.

Edit: not this exact model, but the terms in the description could help! AuKing Mini Projector 2022 Upgraded Portable Video-Projector,55000 Hours Multimedia Home Theater Movie Projector,Compatible with Full HD 1080P HDMI,VGA,USB,AV,Laptop,Smartphone https://www.amazon.com/dp/B07MTCMHZX/ref=cm_sw_r_cp_api_glt_i_87F60EV943NQGTST30VM

since it's a windows based program (and it's impossible to find a raspberry pi for a decent price in 2022, and trying to get a windows program running in linux is never straightforward), I would go with a windows PC, and my recommend is one of these refurbished dell small form PCs. I don't know much about projectors so I just picked the one amazon recommended.

https://www.amazon.com/Dell-OptiPlex-i5-7600T-Windows-Renewed/dp/B07XLF99BY

https://www.amazon.com/Projector-Video-Projector-Multimedia-Compatible-Smartphone/dp/B07MTCMHZX

AuKing Mini Projector 2021... https://www.amazon.com/dp/B07MTCMHZX?ref=ppx_pop_mob_ap_share

This projector has worked really well for me. It’s fairly cheap also.