So I tried out a newer Final Fantasy III (Famicom) Translation on the NT mini yesterday that changes the game to MMC5, and it worked fine, and saving worked too. https://www.romhacking.net/translations/2701/
However, the save file is created in the root of the SD card, and not the directory that the ROM is in. So when I turned the system off and came back later, there were no saves showing on the game, but once I moved the ROM to the root directory of the SD, it was able to pull up the saves. Figured I'd post that here in case anyone else has this issue.