If the set $S$ is arbitrary, then no strong variant of $(*)$ holds unless $G$ has order $2$.

Indeed, consider a factorization of $G$, and suppose there is a factor $H \sim \mathbb{Z}_n \neq \mathbb{Z}_2$. Let $S$ be the set of all elements of $G$ such that their projection on $H$ is at least $n / 2$. But then $\alpha^+(S) \geq \frac{n - 1}{4n}|G|$ for $A = [0, \lfloor (n - 1) / 4 \rfloor]$. Also, $\omega^+(S) \geq \frac{\max(1, n / 4 - 3 / 2)}{n}|G|$ for $A = [\lceil n / 4 \rceil, \lfloor (n - 1) / 2 \rfloor]$. We can see that both $\alpha^+(S), \omega^+(S) = \Omega(|G|)$, hence LHS of $(*)$ can be $\Omega(|G|^2)$, which is only a constant away from the trivial $|G|^2$ bound.

If $G = \mathbb{Z}_2^k$, then $G$ has order 2, and $(*)$ does hold since addition and subtraction are the same thing.

If the set $S$ is arbitrary, then no strong variant of $(*)$ holds unless $G$ has order $2$.

Indeed, consider a factorization of $G$, and suppose there is a factor $H \sim \mathbb{Z}_n \neq \mathbb{Z}_2$. Let $S$ be the set of all elements of $G$ such that their projection on $H$ is at least $n / 2$. But then $\alpha^+(S) \geq \frac{n - 1}{4n}|G|$ for $A = [0, \lfloor (n - 1) / 4 \rfloor]$. Also, $\omega^+(S) \geq \frac{\max(1, n / 4 - 3 / 2)}{n}|G|$ for $A = [\lceil n / 4 \rceil, \lfloor (n - 1) / 2 \rfloor]$. We can see that both $\alpha^+(S), \omega^+(S) = \Omega(|G|)$, hence LHS of $(*)$ can be $\Omega(|G|^2)$, which is only a constant away from the trivial $|G|^2$ bound.

If $G \sim \mathbb{Z}_2^k$, then $G$ has order 2, and $(*)$ does hold since addition and subtraction are the same thing.