Another update: After trying various images, CD-ROM Bios files, etc, I've narrowed it down to the patch itself. When I run the game un-patched on my Super SD System 3, it saves games normally. However, when I run the patched version, it fails to save any data (Or load saved data made from the un-patched version).
So I guess maybe the English patch is causing a save game bug, or in some way not compatible with how the Super SD System 3 handles saves? <Shrugs>