Hello,
In our case, SD comes only EOW. So when her bday is during the week, her father drives to her city just to see her and give her a present, but she's normally in a rush because her mother plans a dinner with her, her boyfriend, etc. (knowing very well that my DP ALWAYS goes to see SD on her bday). And the following weekend, whether the SD is with us or with her, the BM plans a party (normally with her friends, as SD doesn't really have a lot of friends of her own, she's pretty reserved). So basically, by the time the SD is with us, the bday is long gone. I mean it's not necessarily "bad" as SD has 2 celebrations, but not really with us or even with her dad.
I think that SD (16 as of today) knowing that her father drives 35 min one way to see her could have some input in telling her mother that she'll come at bit later so she can also have a moment with her dad, but it seems that whatever the mother is planning is never questionned. We could have even driven all together (she has a good relationship with me and my sons, especially the little one), but we're not going to drive through the traffic just to see her being in a rush to leave.
I wonder how other people are doing it.