I would suggest dumping the bios into the GBA rom directory via FTP/SAMBA, then SSH into the machine (or connect a keyboard and quit out of EmulationStation), and move from the GBA rom directory to the one you mentioned.
Of the top of my head this should do it I think:
cd into the GBA ROM directory mv gba_bios.bin /home/pi/RetroPie/emulators/gpsp/raspberrypi/
you shouldn’t need root to use the command but just in case just add ‘sudo’ in front of the mv command.
You won’t be able to find the correct directory on another machine (via FTP or SAMBA) as it isn’t shared, the only way is to manually copy it via the RaspberryPi.
Hope this helps!!