$\newcommand\Om\Omega$No. E.g., for natural $n$, 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, $n$.
A probability distribution is a measure. So, you should write $P(\{x\})$ instead of $P(x)$, assuming that the singleton sets $\{x\}$ are in the underlying $\sigma$-algebra -- which, looking at the context of your post, appears to be the largest $\sigma$-algebra over a discrete set $\Om$. However, in the answer above I used your notations, such as $P(x)$. Also, $P_S(x)$ is a number, not a distribution, conditional or not.