DIYing it is probably too painful to be doable. You won't be able to source any kind of protocol translation chip, so you'll have to send it essentially raw into quad SFP+ transceivers. Running 4+ fibers instead of the required 2 (or even 1) is very expensive, and any kind of WDM immediately blows up your budget. Unless you're getting the stuff for free from a DC renovation or something, it's just not worth it.
On top of that you also have to deal with designing board for extremely fast signals, which is pretty much impossible to debug without spending "very nice car" amounts of money on tooling. People have done it before, but I definitely don't envy them.
[0]: https://www.startech.com/en-us/audio-video-products/st121hd2...
[1]: https://www.blackmagicdesign.com/products/miniconverters/tec...
Probably a box on the source end to manage DDC and strip HDCP.
I think many of those chips are simple off-the-shelf parts. Probably you would need special licenses only to decode HDCP.
If you have an FPGA, you could even create valid Ethernet frames and send the data / video stream over any standard switch / media converter as long as you have enough bandwidth and no packet loss. (10G would be enough for FullHD and 25G for 4K if you make it a bit smarter and can strip the blanking interval.)
This is called an FPGA.
The theory being ethernet is such a well developed, easy to source common jelly-bean part that this would trump any gains that specialized transports might otherwise have.
But this is probably just my inner network engineer being disdainful over unfamiliar transport layers.
Failing that, you're probably doing SDI over your own lambda.
"Classic" DVI-derived HDMI would probably be trickier because of variable clock speeds and additional data but modern HDMI 2.1 is pretty similar to DisplayPort in that it uses four lanes at fixed rates and sends data as packets over those.
I would love to be able to use standard widely available fiber patch cables for long distance video runs rather than needing proprietary cables only offered in fixed lengths and equipped with enormous connectors that are not friendly to conduit.
Also these days data rates are getting high enough that even normal lengths are problematic, DisplayPort just recently announced that 3 meter cables will need active components for the full 80 gigabit per second mode, which means that a computer on the floor connecting to a monitor on a standing desk will not be guaranteed to work with passive cables. HDMI also recently announced version 2.2 with a bump from 48 to 96 gigabits per second so they'll presumably be in the same boat.
For most long haul links people still compress, good old h264 or 265 with latencies in the 200-500ms range (plus network propagation), or J2k/JXS and NDI which are more like 50-150ms. Ultimately 200mbit of h265 is far cheaper to transmit than 10ish gbit of 2110, and in many cases the extra 500ms doesn’t matter.