This thread from 2018 discussed a couple of options to stream your Pi's picture to another system, but I don't know how much of it is still viable. As you can see there, I tried to do it myself, but I eventually gave up, because it was either not working or too slow on my Pi 3B. Maybe the latter will be better on a Pi 4.
I also tried a external capture card for around 80€, but although it worked, it added somewhat of half a second of delay, which made most games not pleasant to play.
The reason I wanted to stream the picture rather than just use an hdmi splitter was that my 16:9 / 720p video projector doesn't like the native solution of the 4:3 / 1600x1200 monitor in my arcade cabinet. So I had to convert it somehow, and streaming was one possible way to do that.
Finally, I found a working solution in an external hdmi-to-vga converter, since my projector only has the aforementioned problems only with digital video signals. Luckily, there's no discernable difference in the quality of the analog vga picture, maybe because I spanned the hdmi cable most of the way and plugged the vga converter directly into the projector (it's only the size of a cable's connector without the cable).
Just some experiences that may show you a viable way for your setup.
