What cabbagedad said, plus some.
Personally I would not talk him out of anything, let him work it out on his own. If he's a good athlete the coaches will figure out a way to make it work for them and him. If your son has a real passion for baseball I would Just make the opportunities available to him to work out, forcing him to choose at this age can lead to early retirement from all sports.
When my son was in HS he played football, basketball and baseball all four years. He would do bp at lunch or after basketball practice and sometimes before games. It was more difficult to find time during football, although he played on a Sunday scout team during football. I would guess he did bp 3-5 times a week and threw a couple times a week on average during basketball, probebely 1/2 that during football, although he played in games. I think he also lifted 3 times a week year round.
btw, a position player needs to work out year round, especially on his swing.