My Mophie Air does work perfectly on my iPhone 5S. It worked on my iPhone 5 with iOS 7, as well. Same for my wife's Mophie Helium and her iPhone 5 with iOS 7.
My only suggestion would be to contact Mophie and see what they say.
Well I did and they replied quickly with a mailing label in my email and said "if we see a problem then we'll replace it". Then I asked if I would get my original juicpack back if they don't find an issue and they never replied. I'm not sending my pack in if there's no chance of getting the original or a new one back. This is a little ridiculous that so many people are having issues.