The requirements that they are asking for is already quite draconian. Exchanges providing source of funds seems to be an increasing trend, which nobody should welcome, given the fact that it can be viewed as an intrusion of privacy, and can be used to hold people's coins hostage, essentially.
You could try to negotiate with them, by closing the account and allowing you to remove the funds, but I doubt it'll happen. If they were willing to do that then they would have simply disabled trading, and nothing else.
But yes, some publicity could help but don't count on it, because this could be an obligation that bittrex legally has to perform from a legal standpoint.
I sent them a support ticket and after 8 days (19th june) got a reply from a specialist, asking this
Jeez, they've really not upped their support reply game since the majority of their users jumped ship due to their new KYC requirements for legacy account owners.