Been living with my MIL for almost 3 years, how do I not give a shit what she says/thinks about me?

The last 3 years have had their ups and downs, first year was absolute madness, she would fly off the handle at any small thing. She has since calmed down a fair bit, but the nastiness is still there, Ive tried so many things to get her to change how she communicates, how she treats others, but she is so set in her ways. I tried one last time yesterday and I’m done, heres what happened:

Yesterday she comes storming into the living room asking “Did you take the batteries out of my remote and replace them with old ones?!”. I obviously didnt, and laughed, but dug further because why is this person accusing me of something I obviously didnt do. No logic would get through to her, even when I pointed out that we had gotten the TV we never replaced the batteries and those were old, yet she decided I changed them because they are the same brands of batteries we’ve used in the past, and “batteries dont just die, they slowly dont work”. Trying to prove my innocence was a futile effort because “no innocent person would behave like this”.

She believes I’m a person who makes little white lies all the time (though she couldnt think of an example besides another time when my wife and her couldnt find an ice cream scoop in a drawer, but I did….so I must have hid it and then “found it”). Because I grew up on chaos, and when everything is going well I need to create chaos.

So after learning what she truly thinks of me, and nothing I can say will change her mind. She said “actions speak louder than words, so after a year of me proving myself then she will change her mind”. Despite me pointing out that the examples she gave were not factual, so how could that ever be accomplished.

Sorry for the vent, but this is absolute insanity that I just cant get roped into. I remember a year or two ago I had such a low opinion of her that nothing she said would affect me. But as you can see, I got lulled into trusting her again, and having what she says and thinks about me matters to me.

But how do you actively practice “not giving a fuck” around a person who lives with you (and not be an asshole, because I dont want to create more drama). Reminding myself that she is projecting her issues/baggage on to me helps, but thats all I got right now

PS: Moving out is an option, but not ideal as Im unemployed and have a 3 1/2 year old who she takes care of during the day

TLDR: Lived with my mother in law for 3 years, she thinks I make up white lies all the time, nothing will convince her otherwise. How do I “not give a fuck” when interacting with her and make my life less stressful