I suspect it isn't true as stated unfortunately. If you really need an example, I can try to cook one up.
Probably it is true for something like $P^0$, for a definition see Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems (there are some modifications one can make to that definition too which might make this easier).
Definitely it is true for $P^0$ under suitable positivity assumptions. What can you assume about $M - K_X - \Delta$?.