I just gave that a go and it seems to work fine – I created a game file as you do when starting for the first time, then when the game started I cleanly exited back to EmulationStation and the .srm appeared in the rom folder.
How are you exiting the game?
If its a permission issue check that your permissions allow user “pi” to write to the snes rom directory (unless you are saving your .srm files somewhere else)
See the attached pic.
If you need to update permissions, its probably easiest using a tool like Filezilla.