How to politely encourage academic integrity in non-academics?
Ask for another meeting. Explain politely that you were very concerned by the "bury it" remark.  See what they say.  With a little luck, it'll end there.

If you're not satisfied, you can ask for a letter explaining how they will handle any negative results and make your continued volunteer work contingent on getting a letter that's satisfactory to you.

If you don't get it, walk.  The inefficiencies you're worried about will eventually come to light, and you don't want to be tarred with that brush.

Edit: The remark from Pam to the question has very good advice for conducting the meeting I suggested.  A positive approach early-on is quite likely to lead to a satisfactory conclusion, without need for letters, etc.