The one word I didn’t see in your description of the problem is “content.” Fixing and optimizing the technical components is good – it’s a best practice – but it’s not what gets results. I’m guessing there’s something about the content on these pages that just isn’t good enough to compete for higher spots.
Rankings will always fluctuate, and I’ve typically seen a newly published page that I know was written with a high caliber jump to #8 – fall to #28 – jump to #11 – fall to #36 – and then a few weeks later it’s #6.
I would focus more on dissecting the writing and page structure. Is there a clear heading order on each page? Are paragraphs chunked into understandable segments? Are there relevant internal links? What about external links to back up some of the information presented?
These are the components you want to look closer at IMO.
EDIT: And as far was what to tell the client, let them know that since the technical aspects have been completed, it’s time to take a hard look at the content. You need to inform them that content is ultimately what search engines scrape.