Help me find a budget pico projector

I am looking for a low-cost (around $200 new or used), very small (5" x 5" x 1.5" h or smaller), silent or very quiet, Pico projector for my basement - some of this lloks good - https://bestrobotsguide.com/pocket-projectors Placement options are very limited. I will mount it close to the ceiling with the picture projected lower on the wall, so I need either vertical image controls to lower the projected picture or keystone correction so I can mount the projector tilted down. The screen will only be about 80", maybe smaller. The room is dimly lit (think typical bar lighting), so I don't think I need a ton of lumens, but I do want the picture to be fairly bright.

My main priority is that the projector is small and quiet. I would like it to be 1080p, if possible, but as long as the picture looks good, lower HD resolution is probably fine. I will only use this a few times a year -- party karaoke or "spillover" for superbowl viewing (the hardcore watchers will be upstairs), which is I why I have such a small budget.

I had a mini projector briefly, but fan noise and the size of projector were non-starters for that room. I have no experience with pico projectors, so I am hoping somebody can point me to the best option in the $200ish range (new or used).