I highly doubt Target, nor any other retail would work out any kind of a deal like that, unless it's under warranty.
But better yet, why not just get a replacement battery and fix it yourself? There really shouldn't be too much involved with it. Check out ebay, search for "ipod mini battery". They sell kits that include a replacement battery, the special screwdrivers required, and some even have instructions I think, for DIRT CHEAP compared to buying a new one or getting raped for a repair shop to fix it.
Or heck, I have 6 years of electronics/computer training, and have a bunch of experience repairing stuff like this. Send your ipod to me with 35 bucks and I can get the battery, replace it, and get it back to you, if you don't feel confident in doing it yourself. But it shoudn't be too hard to figure out for yourself if you take your time with it.
http://cgi.ebay.com/Battery-EC003-550mA-for-Apple-iPod-Mini-Generation-new_W0QQitemZ260064071823QQihZ016QQcategoryZ48680QQrdZ1QQcmdZViewItem
Check out that one, comes with a battery, 2 screwdrivers, and instructions for about 10 bucks shipped. Can't beat that.