I don’t think R’ Chaim asks if she was born on Shabbos Shirah and makes an exception in that case. He holds, for some reason, that it’s not a name.
It’s very difficult to understand, because pashtus, there’s no such thing as a name not being a name; if this is what the father named her, even if nobody ever gave that name previously, it should be considered a name. This is how names in Tanach originated, and nobody I’ve spoken to has a mekor for an end date for new names. I don’t know of anyone else who holds this way.