The least expensive true 4k projector is the $5000 Sony. https/www.sony.com/electronics/projector/vpl-vw285es
In that price range the closest to a full 4k projector you can get will have a DLP panel with more pixels than FHD and a mode to enchance them to look like UHD. http/www.projectorcentral.com/4k-dlp-projectors.htm
https/www.bestbuy.com/site/optoma-uhd60-4k-dlp-projector-with-high-dynamic-range-white/5912304.p?skuId=5912304
You have to be careful with so-called 4k projectors - when you look at the details ... many only upscale.
Here's one I've been seriously thinking about ... true UHD 4k ... native resolution is 3840X2160 ... supports HDR (at least HDR10 but ...I don't think Dolby Vision) ... price is ok (I guess) at $1500 ... unfortunately no lens shift - a feature I really like but not a 'deal-breaker'. As I understand it, they can only be pre-ordered at the moment and I'm not sure when exactly you'd get it.