You shouldn't pass any judgemental opinions to your brother about the girlfriends behaviour. I suspect the girlfriend senses your opinion somehow and feels disliked by you and your mother. I suspect she's behaving defensively.
However she is entitled to dislike fee paying schools and be verbal about it. You're entitled to be pro fee paying schools and be verbal about it.
Your brother is old enough to take responsibility for his own career, relationship and political choices. No one made him give up his job. He chose to give up his job as he wasn't fully committed it to it and was more committed to his girlfriend. It's quite common to blame the partner for a blood relatives shortcomings. I know it's easier for you to blame her, then him.
Have you much of a bond with her? Have you spent quality time together? Are you nice to her? Are you jealous of her?
In relation to your mum, you can say to girlfriend 'oh no mums amazing, because against all the odds she bla bla bla' (explain why she's so great). Or next time the girlfriend passes judgment on your mum tell her 'when you get to know her better, you will see things totally differently (and explain why)'
I this boils down to her somehow knowing how you feel and feeling judged and therefore defensive.
Or alternatively being very judgmental about people as she feels a bit rubbish generally (nothing to do with you or mum). Saying that, politics and education is fair game.
I think you need to be more accepting of the girlfriend and your brothers decision to stay with her. It's got nothing to do with you anyway. Potentially you could easily drive a wedge between you all which long term could negatively impact when cousins arrive.