Vic Searle is looking for a small snare drum, no more than 5-7/8"
deep.  If you can't find one why not make one?  This is exactly what
I did when making a reproduction band box for an orchestrion.  You
should be able to buy the metal parts of a snare drum from a drum
manufacturer, and the skins separately.  All you then need to do is
fabricate a wooden hoop of the required diameter and depth.

My snare drum had to be only 2" deep.  The manufacturer will supply
the two main metal hoops that hold the skin in place, the snare and
the tightening screws, and any decent music shop will supply the skins.

Alternatively, buy a tambourine and use that instead.

Regards, Nicholas Simons