Is there a possibility that Cooper Howard went to vault 4?

As the title says. My reasoning is that vault 4 is one of the only vaults Cooper knew when the bombs fell, it was a vault intended for scientist research into effects of radiations on humans and ghoulification was already known pre-war, so maybe he was turned into a ghoul there.

So my pros are: -He needed a place for his daughter. -Barb's vault location is probably a secret.

And cons: -He hates Vault-Tec. -The doors probably were already shut(ting) when the bombs fell. -Some lines he says when explaining ghoulification to Lucy.