Changing the sound box walls is unlikely to affect the instrument's volume. Thinning the walls may help with resonance and feedback to the sympathetic strings, depending on how thick the walls are. The back plate of hand made violins are often tuned to specific resonant frequencies to help the sound of the instrument.
The loudness of a skin head instrument is mostly going to be made at the face of the instrument. String tension, skin weight, skin thickness, and bridge height can affect how much volume the instrument produces. It may be possible that your skin head is thicker than normal, or may have natural thick spots that is affecting the volume.
Since replacing the head is a major operation, some things you might try first is changing out the bridge, selecting one made of a lighter wood. A taller bridge will offer more down pressure which can help with volume also. It may be possible to get larger gauge or heavier tension strings for the instrument as well. The higher tension will transfer more sound into the skin potentially helping with the echo that you are looking for.