Don't you have to get the exact level of noise that will prevent the AI from hiding from P without letting P reconstruct the AI's actions if it does allow itself to be destroyed?

It's not so sensitive. The AI's actions in the box are very hard to detect from the perspective of fifty years, with minimal noise. The AI expanding dangerously across the universe would be easy to detect, even with a lot of noise (if nothing else, because humans would have recorded this and broadcast messages about it).