I didn't use Bittrex for some time now but I was receiving funds there (I was checking through blockexplorers only) but I've seen that they started to require KYC verification? I think that they have no fiat so this involves all crypto currencies regardless of how low the amount is? I didn't want to make a withdrawal request as I was afraid it would trigger something (like closing account etc.)
They have two levels of verification, basic and enhanced. If it is only withdrawal you are concerned about then basic verification allows for withdrawal of up to 3 BTC per day. To get basic verification you need to provide a name and address although they don't ask for any supporting documents. They then ask for a telephone number and send an SMS code to it. So it is pretty easy to get that done. Enhanced verification allows for withdrawal of up to 100 BTC per day but you do need to provide a copy of your ID and a selfie showing you holding it.
If you do the basic verification and also activate 2FA before you withdraw I believe you should be OK.
Edit:
I just remembered something else about the basic verification process. I gave a UK address and it said they were unable to reference that, then it went to SMS phone verification. I'm guessing that may not apply in the US as they are probably able to cross-reference the name and address with public records.
