The app provides a scannable code to enter (and exit) the store. Right underneath the code, you can check off to use your Amazon gift card balance, but then you can't actually use your gift card balance, as indicated by the sentence right below the checkbox. This built in contradiction indicates the app needs some work.
The help pages indicate that if you use the checkout lanes (instead of just walking out), you can pay with an Amazon gift card (but not a gift card balance that is already in the Amazon account). It does not clarify what happens if the value of the gift card you pay with at checkout exceeds the amount of the purchase.