Rob,
Your son is young, he should keep playing two positions, however as noted you need to make sure that it doesn't become too much for your son on his arm. Your son is probably just going into an important developmental stage, so limited throwing time should be a consideration. How far your son goes is dependent upon remaining healthy always keep that in mind.
I have found that often it isn't the player who decides but others for him.
A good example, as mentioned, Robert Stock in his earlier youth was a pitcher/catcher, caught in college, drafted as a catcher but just converted to a pitcher. I understand he wasn't happy but the people that pay his paycheck made that decision. He wasn't hitting well, and no one was going to turn away a good arm.
You tend to see lots of the better catchers be better in offense than defense. The higher you go the more important it is to be able to hit. There are some guys out there that did both in HS/college and seemed to have survived but they were exceptions rather than the rule. It takes a special player to be a catcher as a well as a special one to pitch at the highest level. FWIW, Jason Motte, the closer for the cardinals was a catcher. Sometimes that strong catchers arm gets another shot at doing something else.
Pitchers are more in demand than catchers on any level, pitchers get more scholarship money, more bonus money, unless the catcher is highly ranked out of HS, more money will keep him away from the draft or more money may keep him out of college. You very rarely see top draft picks HS catchers, I think that this is one position that is really hard to be really good at until you mature physically. BTW, we all know that Harper did both, but interestingly enough, he doesn't do either.
If your son enjoys pitching, let him continue, you never know what will happen. If he is a good hitter he can play another position that may not be to hard on the arm as he is developing as a pitcher. In reality , it will be the coach (s) who will decided in the end.
Just a funny story, one of sons teammates was converted to pitcher at Clemson his spohmore year. His folks were sooooo upset, as well as the player. They even considered leaving the program. He had a good bat and fast as lightening in the OF. Last night my husband and I watched him warm up during the televised yankee game in the bullpen in case he had to go in. My husband was wondering if the parents were still upset.
The above story was to let you know that what happens is often not the players decision, so my suggestion is to keep doing what you want until someone tells you otherwise!