If you suspect too much battery drain, measure it and let's go from there. I have attached the procedure below. And yes, this could be related. Now, I have, in the last ten years, had new batteries die twice in the first 2-6 months - it does happen.

As to the light, it is not normal for it to be on all the time. This is the BCM's way of flagging what it thinks is a problem. Given your wing works fine, I am at a loss as to what it thinks is wrong. I'd do a battery drain test and see what the current is and let's go from there.