a half gallon tank, That in itself is a problem,
Your Spring water another,
Small bodies of water also change temps far quicker,
I've seen Betta's in small containers go from 80 degree way down below 60 just overnight and thats way to much difference to fast,
I don't know how many betta's I've had given to me from "Small containers" that was near death with one problem or another. the last one I was told always had a swim bladder issue, He was pretty much dead after living in a small container for 3 months, I didn't think he'd survive until I got him home,
But he did barely, 3 hours after being in a 5 gallon Cycled, heated tank he was moving, responded to my presence and was even trying to eat.. Next week he gets his worming..
No swim bladder issue now either.
Small bodies of water are just not good for Betta's no matter how you go about looking at it..