The most stable time in Napoleanic France was probably 1810 when Napolean declared the King of Rome. It may seem small but declaring yourself King of Rome made you the ruler of all Catholics and the seccssor to the Roman Empire. At the point, I'd say Napoleanic France was the most stable.
Napoleon was never the King of Rome - it was his son, the future Napoleon II, who was declared the King of Rome - and even then it wasn't a real crown or anything. It was a title reserved for the imperial heir, the same way Prince Charles is the Prince of Wales. He doesn't actually rule Wales.
And I think you're exaggerating the importance of the title. 1810 was far past the point where Catholics cared about stuff like that, and way beyond anyone caring about the Roman Empire. If anything, Catholics were more likely to oppose Napoleon thanks to his annexation of the Papal States.
I think the answer depends on your interpretation of 'height'. In
absolute terms, Napoleon's power peaked between the treaty of Schonbrunn and the Russian campaign (annexing Illyria, Latium, expanding Poland, marrying Marie-Louise) - but Spain and Aspern-Essling showed that the foundation was crumbling, and other countries were catching up to him. So Tilsit is also a totally valid answer, if you pay more attention to the
direction of the Empire.