The big question with a 13u player is whether we're talking about a 7th grader or an 8th grader. The age grouping is pretty evenly divided between those two grades.
If your son is an 8th grader, start thinking of him as being not a 13u, but a member of the class of 2018. The reality is that if he wants to make it to college, he has to compete with and against guys in his grade, not in his age grouping. A 7th grader who is a standout at 13u is on track. With a 13u 8th grader, strong performance might indicate strong talent, or it might just indicate that he's further into puberty than the other guys on the field. Bottom line, I'd want my 8th grader playing at the 14u level one way or the other, so that I'd get a more accurate idea of how he stacks up against his contemporaries.
In the middle school years, my thoughts are that the player should be fully on the adult diamond (not the 54/80 intermediate size), and that the player should be playing his summer and fall ball on a team that emphasizes instruction, pays close attention to not over-using pitchers, requires impeccable sportsmanship and behavior both on and off the field, does not countenance parental loudmouthing, and makes being a competitor something that is expected as a matter of course.
Weight lifting is OK but way down my list. At this age, I find that a lot of your huskier youths are struggling with the enlarged diamond and, quite frankly, would be better served to focus on general conditioning (running) and speed training.
Of all the skills involved in baseball, the one that we find is consistently underdeveloped well into the HS years is running speed. Not everyone can be a world class sprinter, but nearly everyone can be faster than they are. There are instructors who can help. This is a great place to start if you are wondering where your budget is best directed.