I have a different usecase. I have an embedded system that sends out HDMI. However, its boot screen is something I want to replace by another HDMI stream (a static image would suffice). I specifically don't want to change anything on the embedded system for a thousand reasons I won't go into. How do I cheaply and robustly do this?
https://www.ti.com/product/TMDS261B
This and an RP2040 to generate the second signal would probably work just fine; see https://learn.adafruit.com/picodvi-arduino-library-video-out... for an example of how to stretch the rp2040 to do HDMI
Probably not the most elegant solution, but there are many HDMI switchers out there that can be talked to using gpio and/or RS-232. You could use one of those and a Raspi, feed the image in from the Raspi, detect the power of your embedded device via GPIO and then switch over to the other HDMI input. Whether that is a feasible solution mostly depends on other requirements, e.g. regarding power consumption or space.
I used a similar solution once in an AV context and it has been running reliably for years now.
If you want something less hacky Blackmagic has the Atem Mini, a good HDMI switcher that can also store multiple stills and can be switched via ethernet (among other things)