To be fair, I wasn't constraining myself to "affordable" FPGA boards. An expensive FPGA board is a one-time expense, but the license for a proprietary HDMI IP core and HDMI membership is at last an order of magnitude larger, and a recurring expense. You can get some fairly high-end FPGAs before needing the paid tier of the Xilinx tools.

As far as memory bandwidth goes for 4K, HDR10 content is about 2 GB/s, regular SDR content is 1.5 GB/s for normal 24 bit color, and if you're running SDR content with chroma subsampling (4:2:0), you're at 750 MB/s.

If we're considering the "affordable" tier of FPGA boards:

An Artix 7 can be found on boards <= $200, and the GTP channels are capable of HDMI 2.0 speeds (6 Gb/s per lane). HDMI uses an 8/10b encoding, so the gearbox setup would be 600 MHz, which is doable on the -3 and -2/-2L speed tiers. You can also use 64 bit DDR3 @ 1066 MT/s (8.5 GB/s) on -3 speed grades and 64 bit DDR3 @ 800 MT/s on -2/-2L speed grades (6.4 GB/s).

You don't necessarily have to be rendering a 4K stream from RAM to want a 4K output. You could be generating the higher resolution content on demand and not be putting that much pressure on the memory. Think tiled graphics, or native resolution overlays over an upscaled input stream.