I have few questions about the second part.
(1) I'm not sure why $\operatorname{Tor}_1^A(M,\bar{A}) \otimes_B B_P=\operatorname{Tor}_1^A(M_P,\bar{A})$.
(2) I think $\bar{A}$ has a free resolution, but I'm not sure if it has finite length.
(3) Why $\operatorname{Tor}_1^A(M_Q,\bar{A})=0$? I guess $W$ is the union of $D(f_i)$ where $f_i$ is in the minimal prime containing $P$, but not in $P$, but I'm not sure. And why we need the condition that $\operatorname{Tor}_1^A(M,\bar{A})$ is a finite $B$-module?
(1) Consider a free resolution $\mathbb F_{\bullet}$ of $M$ as a $B$-module and note that $B_P\otimes_B(F\otimes_A\bar A)\simeq F_P\otimes_A\bar A$.
(2) Finite free resolution means that the modules are free of finite rank.
(3) If you have a finite $B$-module and a prime ideal $P\subset B$ such that $N_P=0$, then the support of $N$ is a closed set (it is $V(\operatorname{Ann}N)$) and $P$ does not belong to the support, so...