Ohhh, what a shitty situation to be in. No one deserves this. My heart goes out to you and your partner, Sassy, and I'm angry on your behalf!
Disclaimer: I'm currently going through a similar version of this in my own life with immediate blood relatives so apologies in advance if this comes off as a bit militant. 
Now, I actually really like Georgia Ma'am's somewhat sneaky (in a good way!) suggestion of going in kind of covertly so as to not put her on the defensive right off. That does strike me as a very Southern tactic. Kill her with kindness!
As for me, I personally would approach it in steps. Depending on the dynamics at play and how hateful she's been lately, I would go with GM's suggestion first, being sure to use plenty of "I" statements (ie - "I feel hurt when blah blah blah" or "I feel disrespected when [...] ") so as to give her the benefit of the doubt and the option to save face if she's the kind of person who would respond to that sort of thing. After that, I would wait a while to see if there's any change in behaviour. If not, I'd go to the brother, explain the situation, and ask him to speak to her. Wait again. If no change in behaviour, then confront both of them at the same time, firmly and leaving little doubt as to how unacceptable this situation all is. If that's ultimately all for nothing, then only you can decide what the next best course of action will be.
If it gets to the point that you're considering cutting her out, I would suggest making it known that you are, in fact, actively
deciding to cut her out (as opposed to just "growing apart") and why you are doing it so as to leave no room for doubt. You want the focus to be on
her behaviour, not your reaction. Your SIL has to know that her shitty actions have consequences. And if they refuse to change, you have to be able to call their bluff. Your presence in their lives is your only real trump card here that I can see (and the ultimate one at that). If they value you as a person, then that will matter to them.
At its crudest, removing yourself from her life explicitly for this reason is like classical conditioning. SIL learns that when she says X, you do Y, and Z results for her (where Z stands in for unpleasant family discord). Do it enough times and she'll get the hint. She may never stop entirely and she may always be homophobic on some level, but at the very least you want a change in behaviour and how she treats you and your partner. Eventually if your family respects you and has your back, she'll learn to ease off at least in your presence if only to minimize the discord in her own life. The key to this is your brother and also your parents. It really depends on how far you want to take this.
One thing to remember is that your relationship with your sibling will likely be your longest relationship in your life. Don't let her ruin that for you, not without a fight. If your brother chooses her side, that's on him, but I wouldn't give her the satisfaction of sitting there and taking it by default. It's one thing for your own family to turn against you in this way. It's quite another for an outsider, as GM said, to turn the family against itself (which may very well be what she's doing or will do eventually). Keep in mind that while she may be causing some of her homophobia to rub off on your brother (because why hasn't he confronted this on his own yet...?), she's most definitely infecting your beloved nephews with her prejudice too, either directly or indirectly. If you don't confront it, it
will get worse. If for nothing else, think of the kidlets involved. They deserve better. What if one of them is gay themselves?
Also, fuck "tacky."
You deserve better than this. Tacky is when she decided to be insulting to your partner. Tacky is when she decided to block you on social media. Tacky was when she decided to quote Duck Dynasty (seriously, that's really fucking tacky). You're just standing up for your self respect and for your partner's dignity and rightful place in the family. There's nothing whatsoever that's tacky about that.
I actually don't think there
is such a thing as a "neutral" side in family matters like this when one side is clearly being a bigoted, hateful douchebag like your sister-in-law is at the moment. If someone isn't condemning the hateful speech or letting you know they support you and not the other person, then they're tacitly giving their endorsement to the hateful douchebag and the bigoted things they're saying/doing. I think it's perfectly reasonable to expect your own family to take sides on this because really, they will
anyway, and you should be able to
count on their support. It's your human rights and your family of origin on the line here. If they won't support you on
this, then what the hell good are they? You're just putting a finer point on it. If not rocking the boat is bought at the cost of your own dignity, then what good is it?
Shame them. Shame the fuck out of them. Make them examine their own relationship with her and the kind of asshattery they're supporting by letting it go unchecked.
You have the moral high ground here. Use it.
You know, she may be disrespecting your partner but from what I understand, she's also disrespecting
you. The thing she's denigrating your partner for is one thing you share with her (and the one thing your partner likely represents for her as the human embodiment of) and so I would take it
very personally if I were in your shoes. This is the side she's showing you to your face. What do you think she's saying about you on the same matter when you're not around?
I say this all as someone who has gone for long periods without talking to several homophobic family members. Sometimes it works, sometimes it doesn't, but I have never regretted standing up for myself. Ultimately you have to do what's right for you.
I would also like to heartily recommend the following
two posts on the Captain Awkward advice column about the whole "not taking sides" thing. They really helped to change my thinking on the matter and I think they're worth reading for anyone who is or has been in this situation.
Best of luck!
