I understand that some may not want to take his attitude seriously, but if you do have someone like this who you would like to reason with, I think the strongest argument derives from Harry Benjamin's hypothesis that the trans condition is best understood as "neurological intersexed".
The point is that even the Pope would not disallow GRS for an intersexed person, but he has ignored that my trans condition is not a psychological or behavioral condition, but a neurological condition that is just as much an intersexed condition as a genital intersexed person.
In other words, the structures in my brain and the structures in my genitals (and chromosomes) are mismatched. We are not remotely able to modify the brain structures to match the body, but we can modify the body structures to match the brain structures.
We routinely correct other body structures which are in some way dysfunctional, the fact that he cannot see the brain structures that are mismatched with the body do not make them any less real. He cannot see the brain structure that makes a child left handed either, but we all now accept that left handedness is centered in brain structures which cannot reasonably be changed. (though at one time left handedness was considered evil and children were forced to try to learn to be right handed)
Within the Pope's theological worldview, this world is "fallen" and not as God designed it. Disease, deformities, and all sorts of things were not by God's design, but a result of the fall. Medicine routinely tries to restore this "brokenness" as much as possible to more closely match the ideal state of creation. GRS is simply the best that can be done to correct this neurological intersexed condition.
Whether you personally agree with this worldview, many of the people in our world do, and it doesn't hurt to consider how you might best reach them on their terms, if you should choose to try.
I have used this with some success with a number of very conservative Christians, but have not had the occasion to speak with the Pope as yet.