I don't realy know if a USDT private key is identical to a BTC private key... I think it will be difficult for you to recover your BTC. I had made a mistake a little like this, but with BCH at Bittrex. I sent BCH to a BTC address. They didn't do anything at that time. It was just after the release of BCH. They only say that they can do something if it was more than 5000 USD. And i suppose they take some fees for the extra work.