I agree with your PI.  As others have stated, the consultant's statement
that s/he is available should be sufficient.  Any restrictions on
consulting time would be the responsibility of the person's home
institution.  If the faculty member at the other institution were really
more of a subcontractor/consortium, then of course you would need the
appropriate institutional signatures and approvals from their home
institution.

Paul Riofski
Department of Stomatology, UCSF