Same issue on heat
not sure dont think it would operate the AC in high and low fan
The reason I changed my AC in my lance was to be able to run on a honda , Ive had several TC`s with polar cubs , but not made anymore so I put a mach 1 power saver model on that is comparable in amp draw
I put a control box and have it run on a wall stat so I could have auto fan mode that ceiling control units dont do , so the fan would shut off , when the compressor does , like yours does ,I dont need 24 hr set points .... and digital is just another thing to go out in my TC , keeping it simple
I think the Coleman 8330-3862 83303862 Digital Thermostat
you posted at first would work for AC only, but its black , seems like a lot isnt available right now -Covid? the white ones are sold out
Arxcel seems to be switching to different style of stats now
https://www.airxcel.com/rv/coleman-mach/thermostats/digital-
thermostats
But you`d need to call and see what would work - they have bluetooth now
This one looks like it would work with your AC also , not furnace
https://www.pplmotorhomes.com/parts/rv-appliances/air-conditioner/air-conditioner-parts-accessories/colemansinglestagewdisplay_69845
Cheapest one available looks like the analog cool only model