It sounds like your SIL is an unhappy person. Happy, busy, living your best life people have better things to do than slag off others. I’d feel sorry for her if she was nice, but since she’s using you as her whipping boy, she doesn’t deserve any empathy.
Usually I’d say let your DH deal with the batshittery of his lot. However if it’s a personal insult to me, or my DC, then I’d deal with it myself as my DH would play middle man. He’s not dealing with it, so you should.
You don’t owe her anything so no need to be nice.
I too have a tricky SIL who thinks I am awful. What she actually means is, she doesn’t like it that I’m happy, have a nice life, am loved, I look OK etc. She slags me off to anyone who’ll listen behind my back, but does not have the guts to do it to my face. She thinks my DH has “changed”. What she means is he no longer panders to her, spends money on her, because he’s married with a mortgage and kids. It’s all my fault of course.
There isn’t anything you can do to change her opinion of you. So, if the hat fits, wear it. That’s my mantra now. Whenever I have to see these people I put my best foot forward, I’m living my best life, I’m cheery and happy.
I’d also send her a text saying “I hear I’m poisonous 😂 🐍. Better watch I don’t slip anything into your tea ☕️. Have a great day 😘”
Or what someone said below about takes one to know one.
If you’ve earned respect, don’t wait around for it, command it.