@Hulme That's pretty much the cost of a new phone
I believe Nokia Support center can redirect you to an authorized repairer, and they will quote you on 180. However, you will be phoneless while it is out for repair.
Alternatively, AT&T is doing a promotion where you can enroll into Asurion Insurance until May 31st. You can do this all online, so no one has to know about your phone's status. I would wait a few days though before making a claim. Asurion can be a bit sketchy though. I was in a similar predicament as you. Got my replacement phone, the battery is still loose, the rattle sounds different, but my battery is much improved (ha).