I'm going through this right now with my '08 at 229K miles.
If you can still get a reading on the dash after disconnecting the battery, your ECT sensor is *FINE*, for now. No need to change a sensor if that's not the problem.
The problem does not have ANYTHING to do with the air conditioner itself. It just gets disabled by the BCM/ECM to reduce engine load because the ECT told it too cold of a temperature for too long, and the computers are programmed to ASSUME in that case that the thermostat is stuck open and the ECT reading is no longer accurate (even if it is). Hook up a scantool and you can see the ECT reading there even if the dash says (--- F). You should also see a P0128 code but only if enough drive cycles have repeated the problem.
Your thermostat is likely stuck open because the gasket cracked and popped out of place, preventing the thermostat from closing all the way. In hot enough weather, my car is happy, blowing ice cold AC (just charged it last week) and showing temps mostly in the 185-195 range. But on a COLD morning like in the 50s, the stuck open thermostat won't allow the car to get up to operating temperature within 15 minutes (staying under 180) and the symptoms start:
temp display goes to (--- F)
AC compressor no longer engages
cooling fan under hood comes on high speed and stays there
Interestingly enough, no check engine light yet. Disconnecting the battery will reset the system to let you drive with cold AC again. But after 15 minutes if it can't get the temp above 180, you'll be right back where you started. Each time you do this, you'll have to reset your radio's clock and you lose dashboard calculated MPG/MPH averages.
AC Delco 131-158 is the CORRECT part number to use for the L61 2.2L engines. There is another AC Delco part that will fit but will not operate correctly especially in mild weather and it may allow for a stuck closed thermostat when it fails and that invites a blown head gasket, cracked head or block, and other engine damage. It's $30 + shipping on RockAuto, $32 and Prime on Amazon and $48-50 at most local parts stores. If you want to gamble your engine to save $20 on a part, I'm not stopping you, but I am looking at you through a narrowed gaze and judging the hell out of you.
Thermostat ordered. Kitty litter pan ready to catch coolant. Dexcool 50/50 purchased. I'll tackle it Friday. Might try and remember to take pictures and post them. Might try and remember to detail under the hood so it doesn't look so nasty. But hey the car is 11 years old so...
I'm just regurgitating everything I've read here, on YouTube, on other forums found through searched, Reddit, and my own experience. Simple problem. Messy to fix, but easy, and much easier the SECOND time once you've done it, as with most all things, ever.