To clarify this a bit more, it sounds like you've bought some 8-bit game emulator from Amazon, that you plug into your TV and can play various old games on. It's a bit jumpy on some of the more modern games, and you've taken it apart and discovered that it's got an RPi 4 2 GB inside and you're wondering about replacing it. Is that all correct?
If it is, then for one thing I'd say almost certainly you wouldn't be able to just put the SD card into a Pi 5. Usually when the card is flashed with an OS, even a gaming one (e.g. RetroPi) then it needs to know what version of the Pi it's destined for and it installs the OS accordingly.
Secondly, without knowing the provenance and general legitimacy of this overall gaming package, I wouldn't like to say that it's a hardware issue that you can solve with a Pi and not a software issue caused by a buggy emulator. Or it might be just the games themselves, how modern is modern? The Pi 5 struggles with GameCube games for instance. But you've said 1980s: even the Pi 1 could do Megadrive (Genesis) games, so we're back it being a software issue, not hardware.
If you've got a Pi4 2 GB in a little hardware case to make it a retro console, a better hobby might be to get your own SD Card and experiment with getting it set up with an emulator yourself. Don't delete the old card, but I do think your solution is more likely to be in changing the software than the hardware.