A C of C from the Paint Supplier will likely not suffice. There are typically specific requirements for paint (Adhesion, Pantone # or Color Req., etc) that will require the supplier to perform and document testing and inspection activities. (They should reference some ASTM or other adhesion testing specifications on the drawing)
In addition, the Tier 2 PPAP will need to include an AAR (Appearance Approval Report) per customer requirements.