Probably not Miller, your first sentence isn’t an apology, it one for someone else’s behavior which you can’t control so it’s meaningless. If you apologize you should apologize for your own actions, that is the honorable way. Now whether or not the honorable way would be accepted as “good enough” I don’t know. I would suggest you say something like “I made a huge mistake when I replied to your brother with what would obviously make him further upset. Without realizing at the time, I put my desire to ‘get one over’ on him above your daughters (and both yours and my sons) need for an as peaceful and fun as possible first birthday party. Truly, I am so sorry to have disrupted her big day, my apologies.”
That might be a little extra sorry or at least more than you’d like it to be but I think it’s pretty solid, and it covers several bases too. That you are sorry for your actions, you should have realized, how you contributed, what price was paid, etc. If the apology I’ve written is not well (or at least neutrally) received, well….that’s another matter for another day
