I highly doubt that the Apple store, or Apple in general is going to replace the device for you because of a Battery failure (Unless it were an iPod 3rd generation or lower)
If you would have the hard drive error showing again (Im assuming you were talking about the folder with the icon) and took it to the store, and claimed that it was happening constantly, and you've formatted it and restored several times in the past week, maybe they could help you out.
Tricky thing with the minis is that, from my understanding, they dont carry many replacments in stock with all the colors. So, if they dont have it they will most likely ship it out for you and ask you for your new location.
Unless you beg and plead and ask them to take another one out of the box, which they have done for me with products before, but never for an iPod or something near as expensive, usually just In-ear headphones. Didnt really need to plead either though.
Well goodluck.