Let me start this by giving you the benefit of the doubt.
Let's assume you've done your research, listened to both sides, checked records and weighed the pros and cons of voting for either Clinton or a Third Party Candidate and you've decided that Donald Trump is the one who deserves your vote. Let me further give you the benefit of the doubt by assuming that the LGBTQ+ people in your life matter to you enough that you're going to explain to them exactly why, in this historic election, you're making the choice you're making. If this is the case, you'll no doubt need some help explaining to your friends and family who identify as LGBTQ+ why exactly Donald Trump is more important to you than they are.
Let's get to the point; convincing your LGBTQ+ "loved ones" that a vote for two men who will make their lives hell doesn't mean you don't love them.
First, try explaining to your loved one that their rights really just aren't that important to you. As long as straight, white, rich Christians get to stay in a position of power in America, the rest of America just isn't on your radar. It's not that you're against them having rights, you would have cut them off by now if that were the case! You just don't think that someone should be able to use the bathroom that makes them the most comfortable, because it might make you feel icky. Not to mention the idea that LGBTQ+ people should be able to be secure in the knowledge that their job, housing, and children's education aren't subject to them hiding their orientation or gender identity! Really, aren't LGBTQ+ asking for too much?
On the note of asking too much, maybe a little brainwashing vacation could be good for them! Let's not pretend you don't know at this point that Trump's pasty trashcan of a VP still honestly believes that giving LGBTQ+ people electroshock therapy and making them hate themselves will turn them mystically magically heterosexual and cisgender! It doesn't matter how many medical and mental health professionals have spoken out against conversion therapy and the long-lasting damage it can do to someone, surely Pence is just trying to help poor, confused little queer people find the way back to the warm embrace of heteronormativity.
You could try to explain that you're "not racist, but..."
For this talking point, you can conveniently ignore the fact that people of color who identify as part of the LGBTQ+ community are at far more risk of violence than their white peers in the community. The hateful rhetoric against people of color coming from Trump's bleached anus of a mouth since the first time he called people of Mexican descent rapists makes him even more dangerous for the POC in the LGBTQ+ community. (If you can't refer to POC without the use of slurs, it may make this talking point less effective.)
Or maybe you'd prefer to bring up the old "there are LGBTQ+ people who support Trump" point. Obviously these pocket groups of people who either can't fathom the damage Trump and Pence will do to the lives of LGBTQ+ people in America or have bought into self-hatred make it okay to say you support Trump and love your lesbian cousin! It's not like you're destroying all the progress made in America for LGBTQ+ rights with one bubble on a ballot; right?
If none of this works, it may be time to fall back on the idea that there are "bigger issues." You're right, the loves, livelihood and lives of LGBTQ+ people in American should take a backseat to such pressing issues as being able to un-apologetically grope female-bodied people whenever you feel like it. Or maybe your issue of choice is that POC don't deserve to have the same rights granted to every American; in which case you'll have a hard time separating this from LGBTQ+ rights because, unlike the media seems to think, not every queer person is skinny and able-bodied and caucasian.
Or maybe it's time to stop.
Maybe, it's time to look incredibly hard at the issues that matter to you and admit that the LGBTQ+ people in your life that you claim to love will be irreparably harmed if Donald Trump wins the presidency.
Maybe it's time to reassess if that matters to you or not.