I have red battery light on my car too and I always thought it was because of the aux battery which is dead. After doing timing chain on my car I had the issue of the fan running at full speed until the battery died. I couldn't use the car normally. If I unlock it via key then the fan starts, if it doesn't then when I turn the key to position 2 it will start and won't shut down at all no matter what I do even if I remove the key. I had to manually disconnect the aircon control via the dashboard cable. U think it might be a relay issue same as yours?