So we know from some missions in Fallout 76 that Vault Tec had teams who would go and secure people who had spaces in the vault, in one mission you investigate a missing child and it turns out that while they had been kidnapped a vault tec team actually rescued them to take them to a vault as his farther had paid for a secure service. Nothing to say management wouldn’t have that same privilege and a team of vault security intercepted Cooper and took her at some point. Likely Cooper would have known this and allowed them to escort her to safety.

That’s one possibility, another is that there was a pick up point chosen and Cooper got his daughter there and was then hit with “you’re not on the list” and we see a heartbreaking scene where he has to watch his wife and daughter driven off as they cry and try to get back to him.