Suppose that $f :  \Gamma \to \mathbf{R}$ is a trivial quasi-isomorphism and write $f = \phi + g$ where $\phi \in Hom(\Gamma,\mathbf{R})$ and $g : \Gamma \to \mathbf{R}$ is a bounded function as you did. Here we prove that if 
$$|f(x) + f(y) - f(xy)| \le C,$$
for all $x,y \in \Gamma$, then $g$ is bounded by $C$.

First of all, since $\phi$ is a group homomorphism, we have
$$|g(x) + g(y) - g(xy)|= |f(x) + f(y) - f(xy)|.$$

Suppose that $g$ is bounded by $D$ and that the bound is optimal, namely there exists $x \in \Gamma$ such that $|g(x)| = D$. Since


$$2D - |g(x^2)| \le\left|2|g(x)| - |g(x^2)|\right| \le|2g(x) - g(x^2)| = |f(2x) - f(x^2)| \le C,$$ 
we have $|g(x^2)| \ge 2D - C$. On the other hand $D \ge |g(x^2)|$ by assumption, so $C \ge D$. Therefore $g$ is bounded by $C$.