Answer:
Regardless of the state, under the CIP rule the custodian is acting on behalf of a person who is under a disability (age) and it is the custodian who is considered the customer for the purposes of CIP. It is the custodian's information you must verify, not the child's.
You need the child's name and SSN for the purposes of information reporting, nothing else. Requiring a birth certificate as documentation for a UTMA is outside my experience and, in my opinion, is totally unnecessary.