I have AppleCare.
I made an appointment to go to my local Apple store (where I have a good relationship, due to buying so much gear), and told them what was going on. THEY WERE GENEROUS - they had a "new" (new can mean exactly that, or that a bunch of parts that were fine were salvaged from other phones - bottom line, it was shiny, scracth-free, and it works just fine) iPhone 5 in the back which still had iOS 6 installed. We erased my "current" phone, and - voila! - I was back on 6.
If you have AppleCare, it's totally worth giving it a shot. Sooner rather than later. To be frank, your best tactic might be to say that the phone is having problems, you need a replacement, and, if possible, is there a phone with iOS 6 still on it?
Can't hurt to try.
(and btw, that is just one more example why I ALWAYS buy AppleCare - one of the few warranties out there where you can really get your money's worth - through phone support, etc.)