I will add that under the French law, Alexandre is a legitimate child and that his legal and public recognition by his father also authorizes him to raise the noble titles and the coat of arms which belong to the inheritance of his father. All these noble titles, except the one of Prince of Monaco, are French titles (Ex: Duc de Valentinois, Marquis des Baux...).
Alexandre, as the oldest male descendant, by male primogeniture, of Prince Rainier III and Prince Albert II, is also the future Head of Grimaldi Sovereign House of Monaco.
Actually, I don't think that's true. Most, if not all of Alberts titles are Monegasque. They were passed on via a daughter (Rainier's mum) at least once. As titles have to go via a son, this made them all void. As the Grimaldis kept on carrying them, they are now Monegasque and not French. Not my opion, I just read it somewhere. I have also my doubt about the rest of your story, as it seems a bit too simple. For one: as I recall Albert stated that he was the biological father of Eric/Alexandre. That might not have been formulated like that for nothing. Evil lawyer speak and so on. The same with the "Juridically, Jazmin Grace is the daughter of prince Albert" (Mr. Lacoste in the Figaro last week). That is a sneaky sentence IMO.