He was styled Infante of Spain after his uncle Ferdinand VI gave him the title right after his birth. When he was deprived of his succession rights in 1759 it was established that he would never bear the title of Prince of Asturias but, as a son of the King of Spain, even if Ferdinand hadn't granted him the title, he would have automatically become an infante of Spain after his father's accesion to the Crown.
So he was Duke of Calabria, a Prince of Naples and Sicily ... and an Infante of Spain.
He was left behind in Naples because it was thought that he would be a political embarrasment. They also thought it would be better for his mental and physical health, as in Naples, don Felipe had lived all his life, he had his attendants, doctors and nurses there. So, to sum up, it was thought wiser to live him in his "natural" habit rather than take him to a strange environment where he would not only might have become unhappy, but where he also represented a political embarrasment. I know it sounds tough, but that was how things were done back then!