LSAson went through this same decision last year. Got chosen to attend the Skills USA competition and had an important conference game the same week. He chose baseball.
But the circumstances may have been different. It was his senior year and he was the team's closer and knew they would need him. In fact, that game was a nailbiter and son's time on the mound ended up being key to them defeating the conference leader and clinching a spot in the playoffs. Do you even know if you'll play?
I realize you might be hesitant to ask the coach(that's normal). But this is a real conflict for a good reason. You aren't asking for time off to go on a vacation, it's an honor to the school that you were chosen and you need his help in making your decision. Something you'll learn is that speaking up for something like this and working with him may end up lifting you up in his eyes. Approaching a coach takes guts. You'll be showing him that you have the drive to help the team win AND are representing the school well.
The mission trip is different. It is something outside of you being a student athlete at the high school. It doesn't matter how noble the work is, you are asking for personal time off for a personal interest outside of your school. This might be one of those "pick two out of three" choices. Is baseball your third priority? OK if it is...you just have to realize that you might not be viewed as an important part of the team this year if you make it your third priority. Do you want to play in college? If not(and believe me, that is perfectly acceptable), you probably can put baseball 3rd since you have the rest of the season and next year.