Birthday party on night of Shiva Asar Be-Tammuz

Q: Is it permissible to celebrate a birthday party for a seven year old girl on the night of Shiva Asar Be-Tammuz (a fast day commemorating the beginning of the Babylonian siege which led to the Destruction of Temple)?
A: It is preferable to have it earlier. This is obviously not a time of success. Ha-Rav Moshe Feinstein does not discuss birthday parties on the night of Shiva Asar Be-Tammuz but having a wedding on that night. His conclusion is that he is not thrilled with the idea but he does not forbid it (Shut Igrot Moshe, Orach Chaim 1:168). We therefore as not thrilled with having a birthday party on the night of Shiva Asar Be-Tammuz, but if you really want to, what can we do? I seriously ask: it is really not possible to have it earlier?