Minisforum offer a Mini-ITX board with a 16-core Zen4 AMD CPU soldered on for under $400. The AM5 socket version of that same CPU alone is over 500.
It uses SO-DIMM DDR5 so might be an interesting option in your case.
(Yes, it is a mobile CPU but it has the same amounts of L2/L3 Cache as the AM5 chip, just clocked 300MHz slower)
After commenting i looked up the actual capabilities of the port and it turns out while the port is officially only HDMI 2.0 it actually still supports 120Hz, HDR and VRR anyway. So basically it only doesn't support Display Stream Compression for 144Hz and beyond.
I quickly tested this by connecting my PC running Linux with a RX 6800 to my TV (LG C4). 120Hz, VRR and HDR were all available.
https://store.minisforum.com/products/minisforum-motherboard