Can I say: what a hard decision for your girlfriend.
I adore my husband, but were we divorced and my son asked to have daddy present for a family birthday celebration, I honestly cannot imagine looking into my little boy's eyes and saying "no".
Please remember that for your girlfriend, her children *need* to come before her relationship with anyone else for now. From many mothers' perspective, I think we'd feel that she's making a mature decision, not in favor of her ex-husband, but out of respect for her child, who just wanted to have her whole family there the day of her birthday.
Honestly, from what you describe, this really isn't about you. Of course, your feelings are hurt, and that is reasonable. However, if you take this personally and make this one incident the sword you decide to fall on in your relationship, this is going to be a huge red flag for your girlfriend.
Her kids will *always* come first. At least until they are adults and moved out.
Be prepared: her ex is going to likely receive a lot of what you are considering 'unfair treatment'. You might think that your girlfriend is not being consistent in her parenting of 'who is with whom and when", however, it sounds like you are just beginning a relationship with her. For the girls, what would be confusing is being told that dad can't come and then having you, whom they have spent comparatively little time with, there instead. It sounds like the only reason your girlfriend wouldn't have had her ex there is out of deference to you, at this point, and as I said, not to get personal-- this birthday isn't about you.
And I will say this from experience-- for little kids, birthdays mean a LOT.
It sounds like you really care about this woman and the possibility of having a relationship with the kids. Good. Go forward slowly, remember, you are dating a woman who has an entire family to consider. If you are constantly jealous of her ex and expect him to only be there when he is 'supposed to', it's going to be much, much harder going forward with her.
A counselor (either couples or individually) can help you with some of this. Some of this is just going to take time, too. Stepping aside tonight was kind-- and I know it was hard--- but it was the right thing to do for this time.
ETA: S., I want to add one thing, and I mean this in earnest-- Be GLAD that you are *not* with a woman who would use you as a way to exclude the girls' dad. Be really glad you are not with *that* kind of woman, the ones who will create drama and rub you in her exhusband's face and make you his 'bad guy' and 'the problem' between the two of you men. I've met women who did this and lord, the heartache these women were sowing was unbelievable. Be glad your girlfriend doesn't play that-- for everyone's sake. I hope this is the silver lining you can take away from tonight-- that she's classy enough and mature enough not to make YOU become drama for her entire family.