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 $M - K_X - \Delta$?.

It isn't true as stated unfortunately. For example, take $X$ to be an ordinary elliptic curve, $\Delta = 0$ and $M = 0$. Then $S^0(X, \tau(X) \otimes O(M)) = H^0(X, O_X)$. However, for any effective Cartier $A > 0$ and any $\varepsilon > 0$, we have $S^0(X, \tau(X, \varepsilon A) \otimes O_X(M)) = 0$ (this can be checked easily with a direct computation).

However: 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 $M - K_X - \Delta$?.