My assumption was/is that OP can't exchange the Doge because of fees to withdraw from an exchange. If that's the case, I can exchange it to Bitcoin, send OP his Bitcoin and leave my Bitcoin at Bittrex.
If fee is the factor then it will be a good chance to exchange with you. I think op will take this offer.
send OP his Bitcoin and leave my Bitcoin at Bittrex.
I think op should send dogecoin to you and you will send bitcoin to his/her expected btc address.