Ugh, I totally get where you're coming from with this, I have a brother who does the same thing as far as the slurs, actually, we just had a conversation about it.
I like a lot of the things Jacob presented and sometimes it does really help to have facts on hand too, however a lot of times people just don't care. They don't understand the meaning behind the words and even with an explanation of it, to them, it's still not that big of a deal because the amount of oppression we face, they will never have to experience.
I'm really glad that you do understand that those slurs have meaning, very hurtful origins and I really do wish more people understood that. I'm a firm believer in those who have had slurs used against them in those ways being able to reclaim them however feels right, in fact my trans friends and I (and also at times my mom, although that is under specific circumstances and I'll explain some more why) do use those terms to joke. The difference though, we understand where those words came from, we understand their importance and prevalence which so often is not the case when others use those words. For example, my mom has defended and been there, and in the cases she has used those slurs, it's to reaffirm me (an example of this would be that I received threats that said I was a pathetic excuse for a t* and so my mom texted me that she thought I was a damn cute t).
Today, what I did with my brother was to say "you don't get to use that word, you have no idea where it comes from or the harm that is associated with it. I'd really prefer that you didn't use that in your vocabulary, you aren't trans, and have no idea how that may affect people. People are killed all the time, all over the world for being trans"... Sadly his response was "well, white people die, black people die, so do trans people, whatever, it happens, stop being so dramatic, jeez" he continued but that's the main point. What he doesn't know is that a cis-het person has a 1 out of almost 19,000 chance of being murdered, a trans person's are 1 in 12 (***I have to check this statistic because it was from a while ago and it's not sounding right from more recent research I've done, but it still highlights that this is a real issue. If that is wrong though, there are lots of other easily accessible statistics, it's important to double check them, because although I'm finding where that stat came from, I'm not seeing the 1/12 came from specifically and also, I'm pretty sure the other one is off***). He doesn't know the story of those like Brandon Teena, and we will be watching that movie tonight so I can show him these are real people (By the way, that's from the movie "Boys Don't Cry"). What's really helped him kind of snap into in the past with things such as this is showing him those real life examples, so for queer people I'm thinking about Leelah Alcorn, anyone at the Orlando Pulse Nightclub shooting, or so, so many others. I'm the victim of a hate crime and I don't think he gets that.
Part of this was to say, I'm really glad that you want to others to understand, I completely understand why you would be upset, like Jacob explained, these are people you may know that are at higher risk. At the same time though, I do want you to know that his actions, even if he is your brother are never on you, and if he chooses not to take responsibility, that's not your fault that way.
I really hope you're able to sort this out with him.
You have the power to say "This is not how my story will end".