If you cannot procure filler metal to a MIL-E specification you can order your filler metals directly from the manufacturer. Use AWS A5.01 for your purchasing requirement, all testing has to be per Schedule J with appropriate lot classification, and require certified test reports for each lot delivered. A Typical or C of C ain't going to cut it.
Your purchase order should list every piece of information specified by A5.01 as far as the specification, classification, diameter, length, weight, testing requirements, etc. The testing schedule specified determines or lists the extent of testing and the supporting documentation that is part of the manufacturer's deliverables. If there is no certified test report for each lot, don't accept the filler metal plain and simple. This is not your everyday off the shelf filler metal.
You will have to pay for the testing and most likely the filler metal will be passed through a local distributer. Do not allow the distributer to repackage or break the manufacturer's seal packaging.
Your customer (upper tier) should be auditing your operations, reviewing and approving your procedures for welding, welder training, NDT, and other supporting documentation (such as your purchase orders), how materials are recieved and inventoried, etc., before you go into production. I know, you've been doing it this way or that for thirty years and no one has had a problem yet.
Trust me, do it in accordance with the fabrication documents (TP278, 1688, etc.). You do not want to fail an audit. It is a long road to get reapproved for fabrication once you get shut down. A good part of my business is getting companies back on track after they've had a run in with Uncle Sam. It is never pretty, it is never painless, and it is very expensive!
Best regards - Al