I don't know what "structured cash activity" is (although from the context it sounds bad). But here's my two cents, I hope it's helpful.
"This investigation was initiated due to an alert on an internal monitoring system for possibly structured cash activity." I take this to mean that a red flag was raised because there was some activity observed that might be of the structured cash activity type.
"This investigation was initiated due to an alert on an internal monitoring system for possible structured cash activity." I take this to mean that there might be some structured cash activity going on. In this case we just don't know if there was any activity going on at all.
A very subtle difference! Although I could imagine there could be sentences where it makes more of a difference which you choose.
If structured is something that raises red flags, then your argument seems convincing.
If structured cash activity is something that raises red flags, then I could live with "possible" -- but I don't much care for the other person's explanation. I mean, just because something is the object of a preposition doesn't explain why you'd have to use "possible" instead of "possibly." I think it's better to think about the different meanings, and choose the version that expresses your idea more precisely.
(Speaking of which, if I were the editor, I'd be tempted to rewrite the sentence in a less succinct but easier to understand way.)