Technically, Bittrex can recover your Bitcoins but that would be at a security risk to them as they would have to import the private key for their usdt wallet into another Bitcoin wallet, which threatens the safety of the private key which is why they'd only do that if the amount exceeds $5,000
Only Bittrex can help you because only them have the private key to that address.
If they refuse to help you because your transaction is joy up to their limit, then there's nothing you can do.