Back when "gay marriage" was approved in NY State in 2011, an editorial was published in Time: The Bittersweet Victory: Why Gay Marriage Still Isn't Marriage. Buried in that article was the following paragraph:
Marriage without a church or temple wedding isn't the real thing. Why can some people have all the bells and whistles in the church of their choice but not me? Of course, there have been and will be congregations and churches that allow gay men and lesbians to be married in their midst and to bless those unions, recognizing that God loves them just as much as Governor Andrew Cuomo does. But some rich and influential religious institutions [NB: read this as the Catholic Church] are not only free to continue to reject gay men and women as equal beneficiaries of all aspects of faith but will now also rally their congregants to reject politicians who are willing to abide with this extension of secular civil rights — no matter how much acceptance there is of same-sex marriage elsewhere, no matter how many wedding announcements appear in the New York Times.
Consider this recent quote from the Hildabeast on abortion:
"Yes, we've cut the maternal mortality rate in half, but far too many women are still denied critical access to reproductive health care and safe childbirth, and laws don't count for much if they're not enforced. Rights have to exist in practice — not just on paper. Laws have to be backed up with resources and political will. And deep-seated cultural codes, religious beliefs and structural biases have to be changed,"
Do you think she'd be any different on "gay marriage"?
With the above, you should be able to answer your own question: Would Catholic Priests be required to marry a gay couple?
Given your excellent post, I think it is clear: at some point Catholic priests will be required to marry gay couples. I doubt Muslim Imans will be required to do so.
A blanket exception will be made for them.