Could this trash have been avoided? I'm going to go with... no. The only way I can see of getting trashless mozzarella is to bring a jar around to a neighborhood salumeria and see if they don't mind putting fresh mozz directly into my jar. I don't really think they would go for that, though perhaps I am underestimating them.
What I didn't put in the trash today: little tomato stems (compost), empty ice cream container (recycling).