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.
Bittrex is extremely poor in customer support they always take a lot of money from transaction fees and now I boycotted bittrex because of scam projects.
I wonder if anyone has encountered the same error as me in poloniex.
but the point is this mistake is not coming from the exchange, but from the users itself
i think all exchange will always remind their users, to send only the specified crypto to the wallet
for example if we want to deposit some bitcoin, when you click the address link, you will found this words right?
"Do not deposit any non BTC assets to the above address, otherwise assets can not be recovered."so, always double check your activity in any exchange, such as deposit or withdrawal or anything buddy
by the way, i'm sorry to heard about your issue