The best PoD (meaning one before a large rejection of monarchy to large indifference*) in the late XVIIIth would be during the XVIth century or early XVIIth century with regularity of Estates Generals and increase of Parlementarian power (which, should it be remembered, had a really distinct role).
You'd need to weaken Valois (interestingly enough by having them survive, as it would keep Bourbons as important lords in France as well a rival political force), making the use of EG more systematical to gain legitimacy and support.
More factionalism amongst nobility would probably see an uneasy alliance akin to the IOTL Fronde, but with more chances of success in this context.
Basically nobility and parlementarian elite calling for regular EG as the rule and no longer the exception.
The diversity of Parlements could be as much as an obstacle than a motivation to this evolution, especially with the idea of a common body for all the kingdom, but eventually your goal is to keep them as much as possible while having one permanent EG issued from them.
It wouldn't be a constitutional monarchy, more obviously a parlementarian monarchy while itself far from being akin to IOTL British parlementarian monarchy but as the latter, it could provide the structures to evolve more liberaly.
Apart from that survival of Second Empire might work, especially because it didn't considered itself as a monarchy, but as an imperial republic. That said, it depended a lot on the charismatic possibilities of the emperor, as the whole political mythos was made around him, and may fail due to a different crisis as easily it did in 1870 while it looked for everyone as particularly strong.
*The fall of Second Empire is the typical bad PoD for reasons listed
there