About restored French Monarchy, might they be something like say Brazilian Empire, where King has an executive role?
If say it’s a pre-1871 pod then likely yes, as the monarchists would be more active and involved in French politics. Though Thiers was a prominent Orleanist and took charge of the provisional government in the aftermath of the Franco-Prussian War.
Though if you want an even earlier pod, you could have Emperor Napoleon III assassinated in 1858 by a rogue Cabonari member. Empress Eugenia was gravely injured in otl, and if she also passes away as a result of this more deadly bombing, the second Empire probably collapses.
There would be a Child Emperor on the throne and a power struggle would develop as Bonapartism at this time wasn’t really defined and more or less United different factions under Napoleon III. Plon-Plon would be in a struggle over the regency with the rest of the government, and the Republicans likely become more active.
The Royalists would probably also try to make moves in the chaos. If Henri has a child, he could arrange for a marriage/betrothal with the Orleanists later down the line. France however would probably be internal paralyzed for a while which might lead to another revolution of some sort of civil war that would collapse the Fledgling Second Empire.
This chaos probably would allow for Henri to take the throne, or even Louis-Phillipe’s son as the French Catholic base who rallied under Napoleon III would likely shift to one of them. French industrialists would probably be more amenable to an orleanist restoration favoring stability and a more liberal government.