Replacing Duo-Therm Thermostat with a Digital Thermostat on a 2008 T@B DM T1

Hello!  I have 2008 DM T1 with a Cool Cat heat pump/air conditioner.  I experienced some issues with the thermostat this past Labor Day weekend with consecutive 115 degree temperatures.  With these extreme temperatures, my Cool Cat was having issues cooling the inside of my T@B.  I set my thermostat to 85 or 88 degrees but the air conditioner pump would cycle through every 15 to 20 seconds and then would blow air for about 30 seconds then the pump would go again.  On the outside, it would sound like it was struggling with a vibration sounds.  I tried to analyze what was happening and I noticed that when I set the thermostat to a 22 degrees temp difference between the current outside temp and the thermostat setting and use a fan to blow air on the thermostat, it would cycle normally.  I also noticed that the thermostat is almost touching the outside wall. This was the reason why I pointed a fan onto the thermostat.
This is the reason why I would like to replace the Thermostat with a digital one.  Hoping this would fix what I experienced. Has anyone experienced this and is my thought process in the right direction?  Also what is a good replacement digital thermostat with a heat pump function?

Thank you,
Eric