It's hard not to disturb the betta during this process, especially in something that size. I have a one gallon vase with a betta in it. I take a cup that fits in the neck, and take the water out that way. It takes a little longer than pouring the water, and disturbs the betta less.
My concern with your post, however, is that you say spring water.....are you treating this water with anything before putting it in the vase?