First of all, I'm not sure precisely how plausible this is, but here it goes. The point of divergence here is that Henri Comte De Chambord is out of the picture by 1873 at the latest. With the would-be Henri V dead, as they did historically, most monarchists embrace the Orleanist heir, the comte de Paris, as the new pretender to the French Throne. Phillipe recognizes the tricolor flag, unlike Henri historically, and is restored to the throne by the national assembly.
How long can the restored monarchy last? Can it survive through Phillipe's life time at least? What kind of King would Phillipe be? Since he served in the American Civil War, would his accession to the throne have any impact on Franco-American relations? How does France politically evolution differ here after 1873? How much political power would the Comte de Paris have as King. More generally, how are European politics affected by this turn of events?
How long can the restored monarchy last? Can it survive through Phillipe's life time at least? What kind of King would Phillipe be? Since he served in the American Civil War, would his accession to the throne have any impact on Franco-American relations? How does France politically evolution differ here after 1873? How much political power would the Comte de Paris have as King. More generally, how are European politics affected by this turn of events?