I tend to think of these as forgivable lies seeing as how saying the slightly different truth would then spoil a portion of the story

I'm growing on the idea that we had to abandon our family before entering the vault for some reason. Like, they couldn't accept the baby into the vault so your spouse had to stay behind to protect, and then we come out after the freeze looking for them. So when it sinks in that 200 years are past, our character finally realizes their spouse and child are gone.

And what if, when we're talking to our spouse about one of them staying behind, we can be a huge dike about it? Or we can volunteer to stay behind, and that causes a non-standard game over and ending scene?

Then again, our spouse is with us on the elevator when we see the nuke go off. I don't know if at that point they'd let people into the elevator entrance and then say "wait, no, can't take the baby, you'll have to go back out there."

The SS is not unique in terms of knowledge before the war since non-humans such as Codsworth still hold that information and there are likely other survivors from prewar vaults.

If there are not other survivors from vaults, the question of why the SS is in fact the Sole survivor is interesting. Did Vault-Tec deliberately go out of their way to preserve this one individual? Or is it just a coincidence that this one vault happened to have a cryo-pods and that only this individual took advantage of them? I'd actually enjoy the coincidence backstory, but I doubt it will be written that way.

Since you start the story creating a child, I would not be at all surprised if your genetics (and the genetics of your child) have an important role to play in whatever the Institute is working on.