Since it concerns a specific individual's service in a Louisiana command, this subject should be on the Louisiana message board.

By the law of April 16, 1862, the lower age range was eighteen. Before that time any boy who wanted to volunteer and was accepted by an enrolling officer could enlist.
According to the marker at St Stephens Episcopal Church in Pointe Coupee Parish, Ives was born on Aug 15, 1846. He would've been 14-15 in 1861.

If as you mentioned, evidence shows the boy enrolled in school in 1863, he either had been discharged from military service or never enrolled prior to that time. Since other members of the PCA appear on rolls during early 1862, anyone who belonged at that time should have his name reported. If he had been among battery members left behind at Madrid Bend, at the very least there would be a prisoner-of-war record for him.

At the very least, Batchelor should have applied for a Louisiana pension claiming service in the PCA, or have been listed by other veterans as a member of this command.