$\newcommand\Om\Omega$No. E.g., suppose that $\Om=[n]:=\{1,\dots,n\}$, $S=[n-1]$, $P(x)=\frac1n$ for $x\in\Om$, $Q(x)=\frac1{n^2}$ for $x\in S$, and $Q(n)=1-\frac{n-1}{n^2}$.
Then your conditions hold for $\varepsilon=\frac1n$, $P$ is uniform over $\Om$, but (for large $n$) almost all $Q$-mass is at the one point