Sounds like your thermostat may be going on the blink.
The thermostat connects to the AC with a 4-conductor telephone cord. The outer two conductors are 12v
and ground. The inner two conductors carry serial commands Kinda like a USB port on the computer.
The AC/HP is initially in heat pump mode. The fact that it starts and switches to AC shows that the connection is good and the circuit board in the AC/HP can receive and decode commands.
I would check that both ends of the telephone cable are plugged in firmly. The display going blank could be a loose connection since the 12v
comes from the AC/HP.
If the connections are good, I think the thermostat may have problems. From the fact that it blanks out, I suspect that the rudimentary processor in there may be hanging up.
There is a thermostat reset sequence using the buttons that the Dometic Tech showed me, but I don't remember it and my papers are all in the trailer 10 miles away. Maybe someone else can supply the sequence.