Have you spoken to her about this? You said you said "just us 6" but maybe she doesn't realise how much you just want it to be the 6 of you.
It seems like she just wants to see her boyfriend all the time, maybe she doesn't feel right without him or something? Or peraps she's insecure about her relationship with him, as she can't seem to be without him. My old best friend was like that, she waned to see her boyfriend all the time as they had trust issues, and we drifted apart as a result.
Just talk to her about it, say that you hoped your birthday would have just you 6, as that's who you wanted to celebrate with, so maybe you could have a be-lated birthday celebration with those 6 people. If you make it clear that you just want the 6 of you to do something, and tell her how much it means to you, she shouldn't invite her boyfriend along.