Santo
I have no experience with that specific game, but I connect my Steam Deck quite regularly to a 4k monitor, but running that in 1080p. Some games might depend more on the CPU and can take the bump up from 800p to 1080p just fine. I suggest you playing on the monitor you like the color/brightness the most, regardless of native resolution, locking the framerate at 30Hz and enabling FSR on the side menu. Try 1080p and if you notice a performance issue, go back to 720p.
I played R-Type III on the SNES a couple of weeks ago and it also slows down there! But a great game. Very hard 🤣
Beat’em up? Confused me there.
I hope you got it from somewhere where you can return it. This is not normal and seems like faulty hardware.
I’ve never heard of these games, but I’m now interested!
My solution to games not supporting Steam Deck/Linux is to simply not buy/play it. It’s not much but it’s honest work!