Justification of a CB was added in the expansions, so you won't find it in the vanilla game. Justifying a war is, in most cases, like a 10% discount on Infamy. I almost always have my attempts detected within a month, although I once had one go completely undetected, and on another occasion had it detected on the second day. Since it's a linear discount over time, the usual 2-4 weeks before detection amounts to a 10-20% decrease in Infamy cost.
There are also events which speed up or slow down the process, and a slower rate means more time to get detected. I've FREQUENTLY gotten events to slow the process, and the couple of times I've gotten the event to speed it up included that time where it actually succeeded. The event triggered about 2-3 days before it was scheduled to complete anyway. Overall, the expansions provide significantly more depth and balance to the game, but the war justification mechanic isn't one of the better additions.
I much prefer the system used in EU3, where various CBs included "discounts" to Infamy or modifiers to Prestige or war score, rather than "all or nothing" CBs as in V2. It's ridiculous that you can start a war against a major country over some incident without the neighboring countries minding it one bit, or else having them throw an absolute hissy fit because you didn't have the proper lame excuse.
Note that having your game registered here on the Paradox site would reveal icons in your signature block to show which games and expansions you own. The presence of a V2 icon without the expansions might have allowed a speedier resolution of the problem, as it would have been more obvious that you were running the base game.