If you marry her for the specific intent to avoid her deportation, it is illegal and would not work.

If you marry her because you are in love with her and want to spend the rest of your life with her as a married couple, then you have a shot. Keeping in mind that you will have to prove that you are not just getting married to avoid the deportation.

This lawyer who gave her the two options, was it the same one who was arrested? Even so, I think a different lawyer should be consulted. You want to get more details on what is meant by "fight deportation" before you pay for it. On what grounds would he try to fight it? The Refugee again?. She needs to know the specific path the laywer is going to take to try and fight the deporation before we can really comment on it.

As far as seeing the judge, this again is questionable? Based on the little information you gave, even if she was determine to be out of status since the J-1 Extension and left voluntarily, she would only be barred from re-entry for 3 years. However to me, it sounds like with the extension, the refugee hearing and another interview date (Assuming you have that in writing), she is in a pending state.