One of my miniPCs is just a little N95 and it can easily transcode 4K HDR to 1080p (HDR or tonemapped SDR) to a couple of clients, and with excellent image quality. You could build a nice little server with a modern i3 and 16gigs of ram and it would smash through 4 or 5 high bitrate 4K HDR transcodes just fine.
Is that one transcoding client local to you? or are you trying to stream over the web? if it’s local, put some of the budget to a new player for that screen perhaps?