Ask the coach if you can try pitching with the non-dominant hand in practice. Warming up the late arriving players by playing catch is a good strategy.
Next, offer to help pitch batting practice - to the best hitters - left-handed and right-handed. Some coaches don't like players to pitch BP, but it can work. The coach can throw five pitches to a batter, then let a player throw five pitches. Alternating with coach/player pitching works well in my team practices. Sometimes I have a righty and lefty alternate pitching to give players valuable experience.
Try switch pitching in a scrimmage game. After all, there is nothing to lose and it can be a lot of fun for the team.