Or... just have him have some sense of realpolitik and be less idealistic. I don't mean to change his character altogether, I can't say how determined was he to outright refuse any kind of compromise... couldn't someone in his circle make him see that was the best window the monarchy would ever get? That refusing on principle the tricoleur would leave France with the tricoleur and a republican regime?
Henri was a Bourbon and with that he inherited a good dose of stubbornness. Correct me if I'm wrong, but he would need a vastly different upbringing to let him agree that constitutional monarchy = good. One of his earliest indoctrinators in this regard was Marie Thérèse de France, the former Madame Royal, daughter of the beheaded Louis XVI and Marie Antoinette. I think it was Chateaubriand who originally made an offer of educating him as a monarch for the 19th century, and while the duchesse de Berri (his mom) agreed, she didn't really have much clout with anyone really after the secret of her second marriage came out. So, he was entrusted to the guardianship of Marie Thérèse Charlotte who proceeded to give him Jesuit tutors (and absolutism ahoy!).
Also, even if Chambord/Bordeaux dies, Spain's got enough problems of their own with their own monarchy that I doubt anyone in France (except your archconservatives) are gonna take it seriously. It'll be like the scene in
Ever After where the king and queen of France look at the king and queen of Spain bickering at the failed wedding of their children, and King Francis turns to his wife and says "And I thought I had problems".
A Spanish Borbon capable of ruling France doesn't exist at that point - at least not high in the succession: Alfonso XII and the duke of Seville being after the "pro-Absolutist, ultramontane" D. Carlos.
So, easiest way would be to kill Chambord/Bordeaux and let the Comte de Paris succeed as King Philippe VIII (for the "conservatives") or Louis Philippe II (for the "liberals").