Hi Lina,
I've called a few people in my network to ask about your question, but I haven't had much luck. I called the Jimmy Carter Presidential Library and Museum in Atlanta since it seemed to be a relatively similar institution, but I was told their volunteers are actually required to be U.S. citizens simply because of all the background information that is needed. A few organizations I called just haven't been in this predicament before. Another museum who uses Verified Volunteer for their background checks recommended asking if they have a background check from their home country. If they can provide you with a copy of a relatively recent one, that might be a solution.
I personally have not had this issue, but if this volunteer is very interested in supporting your organization, perhaps they can fill roles that require no contact with children. I know those roles are often limited and the volunteer might interested in sharing their knowledge, but I'm sure you know it opens the institution up for too much liability without a background check. If they are interested in roles that don't require public contact, I would still make sure to contact non-family references and do an interview to see what you can learn about the volunteer.
I hope this helps a little, and that you are able to find a solution that works for your institution!
Samantha
Abroms-Engel Institute for the Visual Arts