Why would a lower outside temp cause your Walk in cooler compressor to cut off until it warms up?

because the refrigerant pressure is lower than

the cut-in pressure setting of the unit at very cold temperatures.

try installing a crankcase heater on your compressor. You may have

an older refrigerant, which has a lower operating pressure.

Replacing the older refrigerant with a newer compatible one will do

the trick also. This will bring the pressure up while the operating

temperature stays the same.