The meal is a bit different - I would send her a LOVELY present to reach her on the actual day, so that she can have it then and not have to wait - you can Skype to see her open it - and then arrange a meal afterwards.
But the graduation? You HAVE to be there.
Seriously, sort out Skype, and don't make her wait a month for her present, if you Skype then both you AND your DH get to see her open it, and give her best wishes on her birthday.
I'm guessing that it is a big thing for her to feel ready to celebrate her birthday again, after whatever happened on her 14th, and she is cross that you aren't able to support her with this next step in her recovery?
I do feel that this possibly is a big step for her, and she possibly feels like she built herself up to this being the first birthday in years that she could enjoy, with her family around her, and it probably feels like a backwards step for that not to happen.
I would try to have a meal with her, on the day, as it obviously IS a big deal to her to take this step of feeling ready to celebrate her birthday again. If it means that your DH has to miss it, then so be it, for your DD's sake, I think.
You could always have ANOTHER meal at a different time, that your DH can attend, as a graduation meal? Or just as a celebration meal?
I do think that this is obviously important to your DD to have you there on her birthday, given what you have said about her MH and an incident on her 14th birthday.