While my experience is that the religious far-right in America is difficult to reason with at all (quite contrary to my Catholic friends where I live, who have been nothing but supportive), and often confuse their political ideology for religious orthodoxy, there are ways to reason within the framework of Catholic/Christian theology, at least if they're interested to even consider other points of view.
(Before I get started, please note that I don't necessarily agree to everything I write below, and I'm not writing this to start a religious debate, but to give the OP some ideas on how to explain to their family.)
Firstly, the determination of gender is traditionally
done by doctors, not theologians. It is a matter of science, not theology, and therefore the Church can't really declare any doctrine about it, since it is not within the domain of faith and morals. For this reason, the Church in the past actually did change the baptismal record of transsexual individuals, post-op, upon notice from their doctors. This was then not seen as a change of sex, but as the person always having been their "new" identity, quite consistent with how we see things. Trans women were even allowed to marry men in the Church - for trans men, the shortcomings of current-day phalloplasty made marriage difficult, since there's a requirement of potency (not fertility) to marry.
This all changed in the early 2000's, when a
document was sent secretly to bishops, changing the discipline. From then, birth records are not to be changed, and gender reassigned people are not given access to the sacraments of marriage and ordination. For someone not familiar with ecclesial documents, the language may seem a bit harsh, but in reality, this document is a lot friendlier than it seems (and yes, the Church has a communication problem...). First off, it says
nothing about reception of communion, penance and the other sacraments not related to state of life (baptism, confirmation, anointing of the sick). That means those sacraments, which are the most important ones theologically, can still be received. Therefore, transitioning is
not seen as entering a state of sin, and the person who transitions is
still in good standing. That may change depending on other things (like entering a secular marriage), but transitioning in itself
does not change one's standing with the Church.It also doesn't really take a position about the person's true gender - this is why we are barred from Catholic marriage and ordination. It is a matter of prudence. The Church is essentially saying "we don't know the gender of these people". It is important to remember that science knew a lot less fourteen years ago than it does now, and the document was sadly influenced by dr. Paul McHugh. This is why I'm convinced this discipline will change - and it is important to be aware that this is a matter of discipline, not doctrine. Doctrine is
always publicly declared, and this was not. The jury is still out.
The most important thing about this document is, however, that (in relation to SRS) "It concludes that the procedure could be morally acceptable in certain extreme cases if a medical probability exists that it will "cure" the patient's internal turmoil." This again is in relation to the Catechism's paragraph 2297, which says that surgery (and other changes to the body) must be done only for "strictly therapeutic medical reasons" - sadly, conservatives often abuse this to condemn transitioning, while it's really meant to condemn torture and penal amputations and so on. But the document I linked in any case acknowledges that SRS is permissible in "extreme cases" (and by that is simply meant "cases where it's the last resort", which pretty much is always true for those of us who choose it...), clearly demonstrating that the conservative line of reasoning is doctrinally wrong, and a misunderstanding of the Catechism. Also, if SRS is morally permissible, then HRT, as a much less drastic treatment, must also be so.
All of this, however, is still within the gender binary. When it comes to non-binary identities, there is an extra challenge, since Catholic teaching traditionally operates with two genders, and only two. This will not change (since it is dogma), but it is important to remember that this teaching is
metaphysical. It means that after the resurrection of the dead, the glorified bodies will be either male or female. However, in our current state, there are intersex people, as we have known for millennia. The traditional Catholic understanding has always been that intersex people are either male or female, but that this can't always be determined.
In the same way, since we now know that brains also are gendered, it is quite possible that some brains are somewhere in between male and female. In these cases, the person would still according to Catholic doctrine be either male or female, but there would be no way of really knowing which. Should they be forced to accept an identity, when that acceptance would be no more than a guess anyways? I say no. Just as some intersex people simply identify as intersex, agender or bigender people are equally justified in identifying as something "in between". It is perfectly consistent with the fact that our current biology is not as neatly ordered as it was according to the genesis narrative, or is going to be after the resurrection, according to eschatology.
That said, when discipline eventually is changed (and I'm convinced it will be) to allow post-op transsexuals to marry or be ordained (the latter is less likely but still possible), it is unlikely that non-binary people will be allowed the same. But that says
nothing about standing, and
nothing about sin. People (Catholics as much as non-Catholics, conservatives as well as liberals) too often confuse impediments to marriage or ordination with condemnation. A century or two ago, someone who had served in the military was not allowed to become a priest. That does not mean the Church condemned military service.
Lastly, most of us believe gender identity (be it binary or not) to be innate, not just taught us by society, but something we're born with. Yes, expressions of gender may change depending on culture (what is feminine now may have been masculine centuries ago), but we feel the way we do not because we choose to, but because we were born this way. This is perfectly consistent with Christian teaching. Amusingly, some conservatives turn into radical feminists when faced with this...
I hope this can help a bit, at least as starting points for them to nuance their views. The problem with many conservative Catholics is that they just
assume that the Church condemns anything that is out of the ordinary, not bothering to actually investigate. Even worse, this is sadly true for many priests and bishops too. But the reality is that there is essentially no doctrine about this, just discipline. Which means the individual believer is free to form their own opinion. But simultaneously, they're
not free to make their view binding on others, especially if it is prohibitive - that would be one of the gravest sins of them all, the same sin of which the Pharisees were guilty.
