I should have added that the most recent FinCEN Q&A on CIP requirements (found HERE) indicates that in such circumstances, you are not required to obtain the TIN for purposes of your CIP (search for the second instance of "taxpayer identification"). Again, you must have a "reasonable belief" that you know who your customer is.

As you've suggested, however, your bank's CIP can require that a TIN be obtained. In that case, the regulation requires that your CIP be adhered to.