A variant of the Stothers-Mason Theorem

No.

Take the curve with equation $z^n = x (x-1)$ for $n$ a large odd number. This is a degree $n$ covering of $\mathbb P^1$, totally ramified over $x=0,x=1$, and $x=\infty$. Those $3$ points are the only places where the valuation of $x$ or $y$ is nonzero. Hence $|S|=3$

Furthermore, the map $z$ is a degree $2$ map to $\mathbb P^1$, which shows that $\gamma=2$

The height of $x$ is the degree of the map $x$ (I presume), and so is equal to $n$. This is unbounded, but $c \gamma |S|= 6 c$ is bounded.

First, to give credit where it is due, this should be called Stother's theorem, or maybe the Stother-Mason theorem, since Stother actually published the result first, at least over $\mathbb C[t]$, and (quoting Wikipedia) Mason "rediscovered it shortly thereafter." (As did I, independently, with a somewhat different proof, but that's another story.)

Second, and more to the point of your question, you might look at the very short proof of Stother's inequality in

The S-unit equation over function fields, Proc. Camb. Philos. Soc. 95 (1984), 3-4.

The proof is an elementary application of the Riemann-Hurwitz formula. It's possible that it can be adapted to give the sort of result that you want.

If your intention is to bound the number of solutions to the unit equation $x+y=1, x,y \in G$, Beukers and Schlikewei (Acta Arithmetica 78(1996), 189-199) proved that it is bounded by $2^{8r + 8}$ where $r$ is the rank of the finitely generated group $G$. Now the $S$-units have rank bounded by $|S|-1$ only modulo constants. I am not sure how to deal with the constants.