Let me reformulate the problem in less fancy terms. Let $\pi(x)\in\overline{\mathbb{Q}}$ be a rational function of degree $d\ge1$. As ACL noted, there is a standard estimate
$$
dh(y) - c_1(\pi) \le h(\pi(y)) \le dh(y) + c_2(\pi), \quad (*)
$$
where it is relatively easy to give explicit formulas for $c_1$ and $c_2$ in terms of $d$ and the coefficients of the polynomials defining $\pi$. However, what you seem to be asking is whether we can give formulas for $c_1$ and $c_2$ that depend only on $d$ and the branch points of $\pi$.

I suspect that the answer is no, for the following reason. For simplicity, lets assume that $\pi(\infty)\ne0$, so I can normalize $\pi$ so that its numerator is monic of degree $d$. A rational function of degree $d$ has $2d-2$ branch points, but with this normalization, it has $2d+1$ free coefficients. So fixing the branch points should give a 3 dimensional family of maps with those branch points. By varying $\pi$ within such a family, the values of $c_1(\pi)$ and $c_2(\pi)$ that come from the usual proof will not be bounded. This makes me suspect that by varying $\pi$ within the family, the inequality $(*)$ will fail if we require $c_1$ and $c_2$ to be fixed.

This isn't a proof, of course, but I hope it suggests where one might look for a negative answer to your second question.