I'd go the iFixit route. The phone is only worth $75-$100 in perfect condition so spending $80+ to have Apple replace the battery is only worth it if you really love the phone and want to continue using it. iFixit has a one year warranty not sure about Apple
https://www.ifixit.com/Store/iPhone/iPhone-5s-Replacement-Battery/IF124-002-4
https://www.ifixit.com/Store/iPhone/iPhone-5s-Replacement-Battery/IF124-002-4