I was hesitant from posting because I don't have any verified information, but since nobody seems to know, I'll offer my two sense.
Agudath Israel may be willing to help with some red tape. They have connections, and they may know how to leverage them. As a plus, they started a Ukraine fund, so you may be able to tap into some of those funds for your center.
Although you specifically asked for Non Government Organizations, I will offer one suggestion from within the US Government. Congressman Chris Smith, whose district includes Lakewood, sits on the House Committee on Foreign Affairs and on the Subcommittee for Global Human Rights. In the past, he has been very involved in getting numerous people out of foreign countries, like Yanky Ostreicher for example. He's facing a harder-than-usual primary challenge this year, and it's very likely that he'll be willing to step in and help if he gets phone calls from his constituents. If there is a specific issue in which a US Congressman can help, post it here and we'll get the Lakewood residents to call his office.