The problem of tests that you demand from the supplier or from a certified laboratory is a function of who your supplier is and if his laboratory is up to standard. I beleive that if you do not trust your supplier then you should not buy from them. Has your supplier certification to any quality standards? Have you audited his site? We have supplied parts to the auto industry with metal inserts and the suppliers COA has been accepted. I work in plastics and buy from the major players, GEP, Bayer, Basf etc, I trust their laboratories and have never had any problems with my customers.
If you have any doubts I suggest that you talk with your SQA and check with him, this though could be a trick question and if you ask he will say "of course".