Emanuela explains in her memoirs the negotiations between the family (mainly herself and some French legitimists) and the palace concerning the inscription and the arms on his tomb. The palace rejected to have any foreign arms or titles on it. So, it was finally decided to have the style Royal and the fleur-de-lys as you have described. But Emanuela made it sure that his second surname (and her own one) would not be on the funerary stone, to avoid her grandson having his second surname on it, too. Well..., no comment.
The flag covering the coffin when his remains arrived in Spain was not used at the funeral, because the legitimists wanted the Royal Arms of France and the palace did not agree. So, no flag or arms on the coffin at the funeral.
So, Alfonso, Duke of Cadiz and his eldest sons are buried in Spain because it was their wish and the family did not oppose to it.
I do find it odd however that his sons opted for the Escorial and not for a French burial of a man they considered to be the King of France.
The last (legitimist) head of the House of France to be buried in France was King Louis XVIII in 1824. Since then they have been buried in Slovenia, Italy, Austria, and Spain. Why start a new tradition now?
The decision about the burial of Jacques Henri's elder son Alphonse (at the Monasterio de las Descalzas Reales in Madrid) seems to have been made by Alphonse's first cousin King Juan Carlos I of Spain. On Alphonse's funerary stone he is styled Royal Highness and Don Alfonso de Borbon. The funerary stone for Alphonse's brother Gonzalve refers to him merely as Don Gonzalo de Borbon y Dampierre. In both cases, however, the funerary stones are decorated with three fleurs-de-lis (undifferenced but not in an heraldic shield).
There is space in the Cathedral of St. Denis next to King Louis XVI and King Louis XVIII, but I think that it might be sometime before it is filled.
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