https://magicmusicvisuals.com/ combined with https://obsproject.com/. And possibly OBS all by itself.

I'm not saying this is the most practical or affordable solution compared to a more specialized product, but if you definitely want a hardware solution, I do know (depending on your HDMI spec requirements) this would be achievable with any FPGA dev board that has HDMI In/Out. I can't say I've worked on this exact solution, but I did some basic development with a Xilinx (Z7?) board years ago that involved overlaying and modifying an incoming HDMI signal. What you're looking for should be achievable by the same means assuming there's no signal issues I'm failing to account for.

I haven't used one myself, but I believe that something like a Blackmagic ATEM Mini should be able to do that (and much more).

https://www.blackmagicdesign.com/products/atemmini

So kind of like the old freeze frame feature on TVs?