I tried to reproduce the issue, but I don't seem to be able to. Using the same emulator (lr-mgba), with or without the BIOS, the save file is loaded correctly.
Can you check if you have enough space available on your USB ? The save file (in-game saves) is stored next to the ROM, it's named <ROMNAME>.srm, if you don't see one, then the emulator cannot save the file.
How are you mounting the USB stick to the system ? Did you follow the docs or did you do it manually ?