I may have a counter-example. Suppose $A$ is $\mathbb{C}[x,t]/(t^2=x+1)$, and let $M$ be $\mathbb{C}\langle \langle x\rangle\rangle $ as an $S$-module. Then, $M\otimes_S T=\mathbb{C}\langle x\rangle $ is given an $A$-module structure by defining the action of $t$ as the multiplication by $\sqrt{x+1}$ where $$\sqrt{x+1}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3-\frac{5}{126}x^4+...$$ But $\sqrt{1+x}$ does not defines an entire function and hence the $A$-module structure does not desend to $M$.

I may have a counter-example. Suppose $A$ is $\mathbb{C}[x,t]/(t^2=x+1)$, and let $M$ be $\mathbb{C}\langle \langle x\rangle\rangle $ as an $S$-module. Then, $M\otimes_S T=\mathbb{C}\langle x\rangle $ is given an $A$-module structure by defining the action of $t$ as the multiplication by $\sqrt{x+1}$ where $$\sqrt{x+1}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3-\frac{5}{128}x^4+...$$ But $\sqrt{1+x}$ does not defines an entire function and hence the $A$-module structure does not desend to $M$.