Here's my two cents worth...
It sounds like your son's 11+ year old "inside" muscles are "in tune" with his 11+ "outside" body. I'd hate to see him lose his control by asking his inside body to try to do something his outside body is not yet physically ready to do.
I don't have a medical degree to support my statement, but I have gone through an eerily similar experience when I first began coaching high school ball. I was that "school team" coach you indirectly speak of. Because of my rush to judgement I almost cost one of my players his career in professional baseball. Thanks to the info given me by a doctor friend, I "learned" quickly and stopped what I was doing...and that kid went on to play Triple A ball in the Oakland A's farm system.
While I'd love to see your son get some experience on the high school mound, common sense tells me otherwise. My suggestion is to let him play another position for his school team this year then finish strong as a pitcher for his Little League club. From what you've said he appears to have been used properly as a pitcher (though I don't know how many pitches he's thrown in each of his outings). However, the fact that he most often is behind the plate on his non-pitching days is of concern to me. I agree with Chris O'Leary that you should have him play another position. Even though Little League doesn't permit stealing, the fact that he is continually returning every pitch thrown to him is definitely asking for trouble.
While the jury is still out on just how much a youth should be pitching (i.e. # of pitches as opposed to # of innings) the fact remains that if he pitches on both teams, he's going from 60' 6"" to Little League distance...and that definitely is a big difference (as Chris also pointed out).
As a coach (and as a dad - been where you are) I think you should err on the side of caution. Remember, your son can (and probably will) throw for his school team next year. I'm sure his school coach is willing to wait, particularly if he knows he's getting an ACE!