I just bought a 2020 Cherokee about 2 months ago and have the exact same problem. I do not even have 1000 miles on it and it has been to the service department 6 times. They said it was a faulty battery and changed it yesterday but at -9 degrees this morning, it did the same thing. Remote start won't work, then when I start it it shuts off and won't start right back, but after a few tries it does, the check engine light comes on and different systems show that they need service on the info display. one being the electronic stability system that shut off and when it did my vehicle felt like a galloping horse and other times like I was pulling something very heavy.
Hi there and welcome to JCC
Sorry to hear about this...
Just to rule this out : can you check the battery connectors, to make sure they are on tight ? Try twisiting them by hand on the posts : if one or both move, they are loose and that could cause problems.
If the connectors are tight and the new battery is good (it should be, but we never know...), then you're possibly looking at a bad connection somewhere : a loose ground, or perhaps something deeper in the bus system...
Dealer has to take care of this. Hopefully they find the cause sooner than later...
Oh and -9F is nothing for these Cherokees. Mine remote starts (no block heater) at -20F.
Keep us posted !