How can you modify a SUBSET-SUM instance so evaluating a set outputs either 0 or 1?
However, I want to know, is it possible to create a new instance T, the same size as the original set S, that any subset in S that evaluates to W, the corresponding subset in T (the numbers taken from the same positions) evaluates to 0? All other subsets in T should evaluate to 1.

No, this is impossible. Consider $S=\{1,2,3\}$ and $W=3$. Our set $T$ should be $\{a,b,c\}$ such that $a+b=0, c=0$ and $a+b+c=1$.