However, I know how hard that can be. I hate my brother's boyfriend. With a passion. I try so hard to be nice to him, but he can tell that I don't like him and actually asked my mom if she knew why.
Is she asking you to do things with brother's GF or just be nice to her when you're at family events? Alcohol may be your friend
Married: 11 years, no kids
The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low, and achieving our mark. -Michelangelo
His mother would like us to include her when the other brother and girlfriend come over for dinner. She hasn't been to our house for almost a year, not because she wasn't invited but because she choose not to come.
First of all, forget about how spoiled the younger brother is--that's not your business. I know it's hard, but like phmh, I've been around long enough to have some perspective. If you let this color your interactions with the brother & gf, it sets you up for failure and ruins any good time that you might have.
You say that your SO's mother wants you to fix things, but if the gf refuses to socialize with you, I don't see how it's your problem. Do you socialize with the family (including the gf) when you're invited to SO's parents' house?
I spent years dealing with the X's obnoxious siblings and at least one brother's obnoxious gf/wife. I found that I could get along with her when we were alone, but throw his brother into the mix, and they were the most toxic couple you could imagine. I avoided every opportunity to socialize with them on our own, but I'd suck it up for family functions (or dysfunctions--a more apt description of the X's family get-togethers )
I wish I had a good answer -- there really isn't one. I've mentioned before that before I found the need for SI, I was on a similar board for people with terrible in-laws (specifically MILs.)
The advice there is usually a version of "his mom, his problem," "If I had to do it again, I'd pick someone who had better parents/stood up to them" and "never JADE to toxic people (justify, argue, defend, explain.)" Too late for the middle one for you, but the first and third may be somewhat helpful?
When I stopped trying to be the peacemaker, my life got infinitely better.