Just to confirm a few things:
- There are no other key fobs present when this issue happens, correct? Multiple key fobs in the vicinity of the car can confuse the BCM.
- Have you tried changing the battery in the key fob?
- Does the light still show even when inserting the key fob into the slot next to the steering wheel?
The fact that the car starts and drives indicates that the key you're using is recognized. Anyway, there are other places you can go to get a new fob besides the dealership. Ace Hardware even does them:
https://www.acehardware.com/automotivekeys
BTW, you had another problem when you first posted to these forums where the car
"Won't drive" as it appeared to be in limp mode after replacing the battery and alternator. Apparently, the car does at least drive now. So, what ended up being the problem in that case?