My engine shed has a small PV panel that keeps the 12V battery charged. It might be 40W. So I can run the air compressor alone, with no battery issues. It's sunny enough here at 5600 ft in AZ so that I can use any retired 12V battery. In essentially float service, an extra 4-6 years is common.
The Gast 4AM can't pull through compression, no even close. Air motors have very low starting torque. Good for not slipping on the flywheel but makes decompression required. In 0F weather (very rare here) , I must unstrap the compressor for first start; it has too much drag when the compressor oil is that cold. I do have a glow plug, a great help for winter starts at my elevation.
Since my Listeroid 6/1-ST-3 was destined for dual use air compression for my shop's 500 gallon air tank, I used an air cylinder for each, rack closer and for a valve lifter. Small ones are dirt cheap and I plumbed them with 1/4 drip irrigation line and slip in fittings. Small 12V solenoid valves and an N-ch Mosfet transistor let the Picaxe 40X2 do the "heavy lifting". Only the Gast 4AM needs a big high flow solenoid valve.
If I was starting over, I'd copy Hugh's setup. Air works great but is annoying for leaks; pilot valves, checkvalves, etc., all leak slightly and have an operating lifespan. The plumbing alone probably cost what Hugh spent, and his setup can be maintained by any good mechanic.