A small correction. I believe Sullivan, not Sullair, took over the Joy portable line.
I took a pretty good oil bath from a Joy shop compressor once when it shut down on overload with the inlet valve wide open. It sounds like your machine may be similarly shutting down due to a fault. The very first thing I would check is the thermal mixing valve, as it will cause the compressor to shut down on over temperature, even if it is cold outside. I would also check for a plugged fuel filter if it dies slowly, or lacks power.
Is this a screw or a vane?
If this is running an old 2-stroke Detroit, make sure there is some provision to cut off the air to the motor, as the compressor end can spin the motor backwards at shutdown and cause a reverse-rotation runaway condition under some circumstances.
I"m not sure what you mean by "cold natured". Every portable compressor should be left to idle for awhile when they are started cold, not just for the usual reasons, but there may be ice in the separator and controls which should be given time to melt.