Look at the prepared invasion targeting opinion modifiers. I believe it's called / close to preparing_to_invade_me. That one's hidden, I think. If it doesn't have hidden = yes or something like that, check out whether there's anything else odd about how it's defined (and report back!).The zeroed opinion modifier does continue to appear in game as a zero, I'll take a look to see if I can't flag that as being a hidden opinion.
Or just try adding 'hidden = yes' to some new test opinion modifier, then apply it via event, then check the personal diplomacy tooltip between the two characters. It's possible the modifier option is named something else, but hidden = yes seems really damn obvious and consistent.
There is the possibility that there is no way to hide it by definition alone but only as an argument (hidden = yes) when applying the opinion modifer. In that case, my assertion that you could hide a hard-coded but zeroed-out opinion modifier would be wrong.