A very interesting thought and nice thinking with 了, but I don't think the sentence is getting quite that specific - I say that because in Chinese "had" (I had a baby) doesn't mean "give birth to" the way it does in English. 

Think of this sentence said to you in this context in English. You're a child, around 5 or 6. Your mom says to you, "I'd have spent more time with you, but we had your little sister." 

Could mean the baby was born very recently, as it needs more care than you do, and you're very young. But it could mean the little sister is 2 or 3.