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She doesn't want to give her kids to the gubmint, but she's happy as a pig in shit to hand their privacy over to strangers on Facebook by name.

Is there a way around this and still be able to travel out of the country with her?

I'm laughing just thinking about that whole sovereign citizen vs. ebil gubmint vs. airport customs thing. However it works out, I'm pretty certain it's NOT going to go the way she imagines it will. "Imagine" being the keyword here, lol.

Is there a way this won't affect her in her later years?

Pretty sure when those years arrive, this woman's question will be far more about how to make her daughter stop cursing her, and then later still, how to force the now fully estranged adult daughter via the Ebil Gubmint court system into allowing Batshit Crazy Sovereign Citizen Granny access to the grandkids.

I feel like I gave him away when I had to get everything on him

No, he's only going to wish she had given him away. To people who didn't force him to live without proper ID in a society that revolves more and more closely on exactly that.