This is how the game works as a cartridge---- you enter your initials, you play through a game, and it saves your win-loss record for the initials. On the Everdrive, with everything I've tried, it's not worked this way.
I just played through a game on Fusion (Using the 1.1 ROM) and had my initials saved, and then reloaded the game to see if the eeprom saves worked. It worked. After playing a game and having my initials saved, I tried:
soft reset: Worked. Initials save still there.
Hard rest/power on: Worked. Initials save still there.
Close Fusion and re-open fusion, reload the ROM: Worked. Initials save still there.