The main difficulty you will have is that your topic is not one on which people will realistically be swayed by a 15-minute presentation. Since your question talks about "the Roman Catholic Church legalizing gay marriage," you are actually speaking about how the Church as a whole or individual parishes either approach performing gay marriages or accepting gay couples who have been married in civil ceremonies; the Roman Catholic Church cannot legislate on whether gay marriage itself is legal in any particular country, although it can campaign vigorously.
Those Roman Catholics who believe that homosexuality is a mortal sin are not going to change their minds as a result of a brief presentation; it is an issue most members of the Church have already thought about, and on which they have strong opinions. Thus your best chance at a genuinely persuasive presentation is to focus on some specific aspect of the debate on which your audience might be more amenable to persuasion.
You might argue, for example, not that the Church should perform same-sex marriages, but that your local parish should welcome rather than excommunicate gay couples who have been married in civil ceremonies. By reframing the argument to focus on some more modest aspect of the issue, you have a thesis which actually has substantially more support in both precedent and Church doctrine, and thus is more likely to be persuasive.
The statements of Pope Francis should be your key arguments in your presentation, as he is an authoritative source for Roman Catholics. While he remains unconditionally opposed to gay marriage, he has two positions which might lend support to the thesis that the Church should be more receptive to homosexuals as individuals. First, he has stated that the Church should be welcoming to gay people, saying:
If they accept the Lord and have goodwill, who am I to judge them? They shouldn't be marginalized. The tendency [to homosexuality] is not the problem ... they're our brothers.
Secondly, he believes that priests should offer support to parents with gay children rather than creating barriers to them. In fact, theologically, he emphasizes that the focus of the Church should be on alleviating poverty and suffering rather than policing bedrooms. While this isn't a ringing endorsement of gay marriage, what it does is allow you to argue that the Church should not exclude gay couples married in civil ceremonies from full participation in the life of the Church.
For this, you have considerable Biblical support. According to the Bible, the only person completely without sin was Jesus, and thus you could argue that even those people who believe practicing homosexuals to be in a state of sin have no grounds on which to exclude gay people from Church life. If one excluded all sinners, the Church would be completely empty, as there is no one completely free of sin. Moreover, as Jesus explicitly stated that he came to "save sinners," an important duty of the Church is outreach to all people. Thus even if you address an audience hostile to gay marriage, you can appeal to the notion that the Church has a better chance of reaching out to gay people if they are included in the full life of the Church rather than condemned and excluded.