You could use a refillable butane container, ( small camping gaz or calor ) , or even possibly run the airbrush directly from the compressor. You will have to arrange air pressure relief and a regulator, plus oil and water traps.
By the time you've added a motor and drive, the result will be quite heavy, noisy and inconvenient. I've been down a somewhat similar route.
I realise this goes against ME "tradition" but instead of buying those bits and pieces for a DIY solution, I'd consider putting the money towards a small, self-contained, oil-free compressor marketed especially for air-brushing. That's what I eventually did, money well spent.