Don't let the people who are trying to make this about them distract you. People who attend someone else's birthday party and make a point of sulking and refusing to sing are usually dead ringers for narcissists. You've already listed her other behaviors which point to that - the grabbing and greediness.
As others have pointed out, your family is making a mistake by thinking your brother's wife deserves enabling and every free pass in the book for having been poor and overweight. "Not guilty by reason of difficult childhood" is a verdict many hope for, but there are equal numbers of us who know that we still have the responsibility to learn how to be kind and considerate. "Treat others the way you would like to be treated" is called the golden rule for a reason. It's simply put and without excuse.
Envy is murderous. Every time a circumstance in Scripture involves envy, an account of murder or attempted murder is recorded. Most instances of attractive girls being bullied to death involve obvious envy.
Your SIL's hatefulness and selfishness is palpable, and at some point you're either going to have to provide a safe haven for your gf away from family gatherings, or your brother is going to have to develop a zero tolerance policy on crappy behavior from his wife. IF YOU DO NOT TAKE ACTION, your gf will likely seek out safety away from you and your family. If you don't want to lose her, follow the best advice here and confront her every time by asking her to clarify her snide comments, and label them as rude. Speak to your parents and your brother, and let them know they need to stop enabling and making excuses for her. She's an adult who can seek help for learning social graces and how to handle feeling poorly about herself and the hand she was dealt. Lashing out and grabbing everything for herself is normal behavior for a toddler. No loving parent tolerates that, and trains the child to stop while showing them the correct behavior. Adults that behave like toddlers can lovingly be told to stop, and told to make better choices.
It's awkward to have to deal with other adults who still need parenting and instruction, but it's deadly to ignore. Your brother and parents need to grow a backbone and learn to speak the truth in love. Don't be passive in the face of hatefulness. Get outside counsel for you and your family if necessary, before you lose everything.