As noted above, this is not a CIP issue since a Sole Prop is not a distinct entity - but rather an "aka" of an individual. Once you have performed CIP on the individual, your regulatory obligation is done. Any further verification would be a matter of your own bank policy and prudent banking practices.
If you live in a state that does not require a fictitious business name filing, then you should check with legal counsel that is familiar with banking issues as to a best practice.
FYI - here is a state-by-state list of filing requirements:http://dbafilingonline.com/dba-by-state/