I don't think your anger is misplaced, it sounds like you're not exactly enamoured by your brother's standpoint on this and have already resigned yourself to his shittiness.
But you make it sound like SIL does less than you do, and gets paid for what little she does.
I'm guessing it's not that you think you should receive money if she does, but that you don't think your mum should have to part with money because she's family.
And different people will have different opinions on that.
Does anyone help you at home? And does your SIL have a job, does her helping your mum impact on her job at all?
PPs are right that it's her money and her decision and I don't think you should get involved but I would be miffed in your position, and I could definitely see the same happening with my own family re brother not helping. But that's the fault of my parents, well, father, for demonstrating what is "wifework" to my brother, who went into fatherhood not intending to do nappies etc because our dad didn't huge eyeroll
Don't know what else to say.