Joe is a decent chap, but so are most of the people in DC, and they all have clay feet. I get tired of Catholic politicians selling themselves as Catholics while openly flouting Church teaching. I myself would not perform a gay wedding, or a wedding of divorced and not annulled people, or really any wedding at all because I think that’s a job for a priest, and I would not take a judicial job that required me to marry people or grant licenses. Joe’s job description did not require him to officiate at weddings. I don’t even see Father Martin actually performing gay weddings.
The reason I think the OP might be able to get approval from his priest is he could frame this in a manner that he is not giving any appearance of marrying two people, and certainly not actually marrying them.