That's a complication that, if an error is introduced, could seriously scramble your database. Thus, it is best NOT to do such an update of the field used as your primary key in a relationship. What problem would such a change solve for you?
And I'd think that you'd need the field that auto-enters the serial number in your Client table should auto-enter that value on Record Creation rather than commit to more smoothly facilitate linking a new policy record to a new client record that might not yet have been committed.
Best thing is not to worry about gaps in the masterkey. Unique ID's are just that. Unique. Once you delete the record, you delete the unique masterkey. It's gone.