Oh my gosh. This is what upset me the most about the immigration matter. That people who are trying to go through channels and do the right thing are given such a hard time. And then because of those restrictions (which apparently don't matter when people come here illegally), they can't be married in the church.
They could get married in a civil service and get married in the church later. Has your cousin contacted her Congressmen? They are pretty useless but we have a friend here who had a similiar situation and her friend (now husband) was coming from Ireland and Sen. Strom Thurmond worked magic.
A despensation from the Bishop was denied. So they will be married in a Civil ceremony. Hopefully, someday they will have their marriage blessed by the church. Right now they are so angry and bitter, I don't think they will pursue it right away.
I don't know if they asked for an extension from a CongressCritter. I will suggest calling Senator Grassley. He seems to have the ability to get their attention.