The RadioCrap Mini Audio Amplifier uses an LM386 audio amp IC. Its output is only 0.45W at clipping into 8 ohms when the 9V battery is new. The 9.6V battery will not make it any louder, the huge battery will just last a lot longer before it needs charging.
You want to use a couple of 9.6V batteries which is 19.2V? Then the poor little LM386 will blow up because its absolute max allowed voltage is only 12V.
With a 12V supply the output is only 0.53W at clipping into 8 ohms which is only slightly louder that the 0.45W with 9V.
If you use a car radio amplifier IC like a TDA7240A with a 19.2V supply, then its output will be 18W at clipping into 8 ohms. With a 9.6V supply its output will be about 3W at clipping into 8 ohms.
The schematic and a pcb design are in its datasheet.