I have always had this question as well, and I recently found an answer:)
In Yabia Omer Chelek Even Haezer 4:9 אות ד Rav Ovadia discusses how the custom of breaking the cup at the wedding is not bal tashchit since there is a purpose for it and bal tashchit doesnt apply to that minhag, since whenever we have a reason for the action, bal tashchit doesn't apply. The same thing he writes in Yechaveh Daat 5:46 where he discusses if cutting branches of a tree that produces fruit for the schach is bal tashchit. He answers that it is not bal tashchit because again it has a reason, and it is a mitzvah, and therefore bal tashchit doesn't apply in those situations.
I would say the same thing here: Since the Torah tells us to get rid of the chametz and therefore we have a reason, bal tashchit doesn't apply in this case.