If there were no investigation, they couldn't legally use that FOIA rejection reason. They'd have to give a different reason.
FOIA would be abused like crazy to unearth classified information if this method of questioning worked
Denials for outright false reasons are the norm, responses routinely take 10x longer than legally allowed. Just go browse muckrock for a while to see what a mess it is.
For example, look at some of these requests pending (since 2019) with various agencies: https://www.muckrock.com/foi/multirequest/foia-outliers-queu...
Most of the replies are clownish at best.