Handbook 2,
20.2, addresses the naming and blessing of children. There is no mention of anyone other than parents or guardians there. Handbook 1,
16.2, provides more detailed instruction. Section 16.2.4 specifically addresses naming and blessing of children when at least one parent is not a member and the permission required from them.
Assuming both parents have given verbal approval as required in HB 1, 16.2, then the child becomes a member of record after the membership record is created. The membership record for a member of record should be in the ward in which they physically reside (see HB 1,
13.6). First Presidency approval is required for exceptions.