P.M.
Telling a child to stop melting down usually has exactly the opposite effect. But swooping in quickly to empathize: "Oh, my, you missed the start of the song! You feel sad and wish everybody can do it again! Well, sweetie, let's sing it again together, okay? Others can sing along if they want to…"
With a little practice, you can probably get pretty good at making molehills out of his mountains. Also, find many opportunities to talk with him when he's not upset, and give him alternative language to express his needs and wishes. Role-play, with him and his stuffed animals, different social scenarios. Often kids get frustrated when they can't express themselves clearly, or in a way that gets them heard by the people in charge. When he has some practice realizing he has alternatives, he will probably start using them.
Good luck. Sensitive kids do bring their own special challenges, whether these are genetic or learned.