In my opinion, your son is 11. While I certainly cannot and will not tell you how to raise your child, as it is not my place, I will give you advice from my personal experience. I know when I was 11, I was not concerned about throwing long toss in the offseason. I was worried about impersonating Kobe Bryant in the driveway. Playing Mario kart with my friends, and a variety of other 11-year-old things. I'd say your son should enjoy being 11. I wouldn't worry about specialization yet. I'd let him be an 11 year old. You obviously don't want to burn him out on baseball.
One thing is for certain: he does need time off. 3 months of no throwing is great! Regardless of what you do decide, time off is a non-negotiable.