I'm experiencing the same problem with Rhythm Heaven. All my other games save just fine. Any light to this issue would be greatly appreciated.
EDIT:
(I'm assuming you have a G6 Real like me.)
Alright, seems like I got it to work after fiddling around. Select Rhythm Heaven on your G6 Real like how you'd do normally and it should bring you to its Config screen. For the option "RT Save/SoftReset", select "Reset (L+R+B+A+X)." Now go ahead and start the game. After you see it finish "Saving", go ahead and press L+R+B+A+X, and it should bring you back to the G6 start up menu. Just start Rhythm Heaven normally, but this time, no need to change anything on its Config screen, and you should have the save file.