Hi ,
In this scenario your copy control only copies the partner function available in Delivery and sales order. it does not redetermine here . so how come it will fetch the new contact person in billing document.
So better you need to assign in in sales order and copy from there to delivery and Invoice . else you can do assign a new routine in copy control for checking the partner function from customer master in invoice . but it may harm in business where ever many partner function mentions in customer master. Need so much testing.
So check you requirement and do the setting .
Hope it will help you.
Regards
Pitabash