Sir!! You are a genius .
I initially figured out the mystery BTC address might come from doge's PK.
Hey thanks for sending me my BTC back.
They can't recover the BTC sent to that addy, only you can do it, if you don't know how ask for help, and ingore any PMs from people who tell you they will help you recover your coins, any advice given must come from a trusted person on forum, the process is rather simple you can do it yourself, just follow what HCP did, if things are still unclear, ask here and members including myself will teach you how to do it.
After getting your money back, you should return the money they sent you, at least this is what I would do if I was in your place because no matter how you look it at, it's stealing!, you should inform them and if they decide that you should keep the money they sent for the inconvenience they caused you, then keep it, if they ask you to return it, please do.
Ofcourse i would send back what they gave me, but i do not have the doge pk to even try and recover the btc.