I'd suggest having a good chat with your lab's manager, and probably with some of the alternatives, to understand exactly what they can offer for your raw material types, what the limitations and strengths are for the methods, and whether they can suggest follow-up testing to understand the scale of contamination / adulteration where applicable.
As Scampi noted, for some commodities you're always going to get some degree of presence of other things, and modern analytical methods are very sensitive so potentially will detect these.
Your suppliers might be willing to amend their specs to mention known potential "contaminant" materials, or they may well have a position statement or similar on them, either of their own or from an industry body.
As for "low-cost" testing options, this isn't always something that exists, so you may need to explore other potential avenues for verification. I have materials for which the only useful analytical approach to authenticity starts at about £500 per sample, and if this finds anything requiring further investigation then the follow-up is over £2000 per sample. Testing every batch of this would make me very unpopular with people who need to worry about boring things like money and profit margins 
In terms of response in the event of a failure / positive detection of a potential adulterant, my process very much depends on exactly what has been found. Except for the methods that give quantitative results in a format that I can useful interpret, my first action is always to call the lab and have a chat with them first. I might be able to get some "off the record" (non-accredited...) interpretation, or agree sensible options to follow up with further more targeted analysis.
If I can't make useful semi-quantitive inferences from the results, that have validity in the context of the material, then I'm unlikely to raise it with the supplier beyond a cursory "we've found this and we're looking into it, so please provide your initial comments" type of thing.
The response varies widely, from the good suppliers who give intelligent and reasoned feedback, through to those who have outright denied anything is wrong even when presented with solid evidence that the product they've shipped has been grossly adulterated to the extent that circa 50% of the material is the adulterant rather than the intended product.
N.B. You get bonus points if they change story part-way through the investigation, from "nothing wrong" to "we accidentally shipped you a different product", only to then send you an even more heavily adulterated product when you agree to take one more trial order for some development work, presumably not expecting you to test it despite having just sent back the previous material... (yes, I've seen this happen)
If you're able to then talk to a contact at the supplier by telephone first - it makes it much easier to come across as a collaborative enquiry rather than an accusation, as you never know quite how someone is going to read the tone of an email. Sometimes you'll get to talk to someone who is extremely knowledgeable and come away feeling more reassured about the supplier and the material, and be in a better place to look at monitoring the authenticity of that type of product. Other times that won't happen, and you'll end up in an acrimonious exchange of solicitors' letters about who owes what to whom and when, but it's usually worth a try IMO.
