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One answer is that there are NO rational expressions in $P_1, \dots, P_{2d-2}$ which that allow you to single out one of the functions $\phi_i$, or every that even allow you to single out any a nonempty, proper subset of the set of all $\rho(d)$ functions. The proof of this is the usual argument of this type (also in Harris's paper on Galois groups of enumerative problems). The parameter space of degree $d$, $1$ dimensional linear systems on $\mathbb{P}^1$ is irreducible; it is just the Grassmannian of $2$-dimensional linear subspaces of $H^0(\mathbb{P}^1,\mathcal{O}(d))$. There is a dense, open subset $U$ parameterizing base point free linear systems with $2d-2$ distinct ramification points. As a dense open in an irreducible variety, also $U$ is irreducible. There is a morphism $f:U\to \text{Sym}^{2d-2}(\mathbb{P}^1)$, i.e., $f:U\to \mathbb{P}^{2d-2}$, sending such a linear system to the corresponding ramification divisor.

A rational expression in $P_1,\dots,P_{2d-2}$ is the same thing as a rational transformation defined on a dense open subset of $\mathbb{P}^{2d-2}$. So a rational expression choosing one $\phi_i$ would be the same thing as a rational section $s$ of $f$. Since $f$ is quasi-finite, the image of $s$ would be an irreducible component of $U$. As $U$ is irreducible, there can be no such rational section. More generally, a rational expression singling out a nonempty proper subset of size $\sigma$ is the same thing as a union of irreducible components of the domain which has degree $\sigma$ over the target. Once again, since $U$ is irreducible, this can only occur when $\sigma$ equals $\rho(d)$.

Of course you did not specify that you want your rule to be a "rational expression" in $P_1,\dots,P_{2d-2}$. Since $f$ is finite, \'etale over a dense open subset of $\mathbb{P}^{2d-2}$, obviously there do exist \'etale local sections, i.e., there do locally exist "algebraic" functions which pick out some $\phi_i$. I have not heard of any nice expression for such (e.g., in terms of other special functions such as modular functions).
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One answer is that there are NO rational expressions in $P_1, \dots, P_{2d-2}$ which allow you to single out one of the functions $\phi_i$, or every allow you to single out any nonempty, proper subset of the set of all $\rho(d)$ functions. The proof of this is the usual argument of this type (also in Harris's paper on Galois groups of enumerative problems). The parameter space of degree $d$, $1$ dimensional linear systems on $\mathbb{P}^1$ is irreducible; just the Grassmannian of $2$-dimensional linear subspaces of $H^0(\mathbb{P}^1,\mathcal{O}(d))$. There is a dense, open subset $U$ parameterizing linear systems with $2d-2$ distinct ramification points. As a dense open in an irreducible variety, also $U$ is irreducible. There is a morphism $f:U\to \text{Sym}^{2d-2}(\mathbb{P}^1)$, i.e., $f:U\to \mathbb{P}^{2d-2}$, sending such a linear system to the corresponding ramification divisor.

A rational expression in $P_1,\dots,P_{2d-2}$ is the same thing as a rational transformation defined on a dense open subset of $\mathbb{P}^{2d-2}$. So a rational expression choosing one $\phi_i$ would be the same thing as a rational section $s$ of $f$. Since $f$ is quasi-finite, the image of $s$ would be an irreducible component of $U$. As $U$ is irreducible, there can be no such rational section. More generally, a rational expression singling out a nonempty proper subset of size $\sigma$ is the same thing as a union of irreducible components of the domain which has degree $\sigma$ over the target. Once again, since $U$ is irreducible, this can only occur when $\sigma$ equals $\rho(d)$.

Of course you did not specify that you want your rule to be a "rational expression" in $P_1,\dots,P_{2d-2}$. Since $f$ is finite, \'etale over a dense open subset of $\mathbb{P}^{2d-2}$, obviously there do exist \'etale local sections, i.e., there do locally exist "algebraic" functions which pick out some $\phi_i$. I have not heard of any nice expression for such (e.g., in terms of other special functions such as modular functions).