I opened up the case of the little medical compressor today, it is a diaphragm type. The diaphragm is 2 5/16" diameter, by about 1/16" thick. The stroke is 1/8", the crank is formed on the end of the motor shaft. the motor is a 1450 rpm shaded pole type, could not see the specs, probably between 150 and 200 Watts, and that's about it.
A suitable compressor would (in my opinion) have a similar motor, a diaphragm about 3"/3 1/2" diameter, a stroke of around 3/16". I'm not sure what sort of valve is in the pump, probably a reed valve. Ian S C