OK to hit on Pitching "Rest Day" or best to completely rest?

My son is a catcher who also gets put in to pitch when we have longer tournaments (2-3 day.) Our coach is mindful of arm health and will have him only catch one game if he might pitch that day (he also plays IF), but it still ends up being a lot of throws, so we are always sure to rest his arm after a big weekend. My question is, the day after a longer tournament when he has been throwing a lot, is it still OK to hit, or would it be better to completely take a day off? We don't have practice Mondays for kids to rest, but we have a cage in our backyard and my son tends to still want to hit when he has a day off. Is this OK, or should I encourage him to completely rest? He is 13 and plays 13U travel ball. Thanks!