He needs to be 40 lbs. before moving to a booster, but that is a bare minimum. Since he is not even 5 (or mature, you say), I would not booster him any time soon.
If he is below 40 lbs. and the straps are ABOVE his shoulders (the height regulations don't mean much, you need to SEE where the straps are), he can stay in the smaller harnessed seat. The weight, though, you need to follow to the exact pound.
So, again, ignore the height limit and see if he REALLY fits, meaning the straps come from above his shoulders, or are at it. If they are below, he cannot be in that seat anymore. A larger harnessed seat would be great until age 6 or more.