Does the sync LED on the OSSC stay red, or does it sync back up? If it does come back, then what’s happening is a resolution switch. A lot of games of that generation used 240p for gameplay but 480i for menus. When using analogue video on a CRT, the switch is seamless; however, resolution switches over HDMI require devices to disconnect and redo a handshake, and this procedure can take a few seconds.
If you’re fine with targeting 480p for both 240p and 480i sources, you can try an EDID minder between the OSSC and your display; otherwise, there’s not much you can do beyond tracking down and paying exorbitant amounts of money for old video processors, like those by DVDO or TVOne, that can handle the resolution switches and scaling to 1080p.