Given a natural number $n$, is there always a rational function $f(x)=\frac{g(x)}{h(x)}\in \mathbb{Q}(x)$ of degree $n$ (the degree here being defined as the maximum of numerator and denominator degree) all of whose critical points are simple and $\mathbb{Q}$-rational (in other words, the numerator of the derivative is a separable polynomial of degree $2n-2$, totally split over $\mathbb{Q}$)?
The analog for polynomials instead of rational functions (when ignoring the multiple critical point at infinity) is of course obvious - one can just start with any totally split polynomial and take the anti-derivative, but this does not carry over since the anti-derivative of a rational function is rarely rational.
Ideally, I would like the critical points of such a function to map to pairwise distinct values under $f$, but since this is a "generic" condition, it might just come automatically out of having a "sufficiently rich" family of examples. This is motivated by some problems in inverse Galois theory, the condition that critical values are simple and map to distinct points meaning that the function defines a cover $\mathbb{P}^1\to \mathbb{P}^1$ over $\mathbb{Q}$ with (Galois group $S_n$ and) all inertia groups generated by transpositions (a very common situation), and the extra splitting condition being presumably very useful for analyzing the local behavior in fibers of this cover.
EDIT (copied from the comments): $f(x)=\dfrac{x^2(x^3+x^2-20x-72)}{58x^3-514x^2+243x-69}$ gives an example in degree $5$, with the eight critical points $0$, $\infty$, $\pm 1$, $\pm 3$, $12$, and $\frac{23}{29}$ all mapping to distinct rational values. I don't know of any examples in higher degree.
EDIT2: For degree $6$, Peter Müller gives the following example (I've only applied some parameter changes to make the branch locus symmetric around $0$):
$$f(x) = \dfrac{x^6 + 53x^4 - 5940x^2 + 62208}{x(3x^4-172x^2+1600)},$$
which is particularly striking because this shape shows that the cover $x\mapsto t:=f(x)$ was obtained by pulling back the cover $x\mapsto s:=\frac{(x^3 + 53x^2 - 5940x + 62208)^2}{x(3x^2-172x+1600)^2}$ along $s=t^2$. The latter cover $x\mapsto s$ has critical values $0$, $\infty$, and five more which are all squares (hence the pullback being possible with all critical values rational)! Is there any explanation why this was possible? In particular, does this generalize to higher degree?
