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Let $G=(V,E), Esubseteq V^2$ be a finite directed graph. For $ninmathbb{N}$, consider
$$chi_{G}^{leqslant}(n)=#Big{f : V to {1, ldots, n} Big| big(forall (u,v)in Ebig)big(f(u) leqslant f(v)big)Big}.$$
It can be shown that this is a polynomial in $n$ of degree $leqslant|V|$ with rational coefficients.

We could take $<$ or $neq$ in place of $leqslant$. Thus $chi_{G}^{neq}$ is the chromatic polynomial of $G$.

Do $chi_{G}^{leqslant}$ and/or $chi_{G}^{<}$ have a name? How to compute $chi_{G}^{leqslant}$ for moderate-sized $G$?

For computing, something similar to this would be sufficient to me. I don't see it myself yet. Something along these lines might also help, but currently I'm feeling stuck.

For extending deletion-contraction: even if we assume $G$ pruned (with cycles contracted, loops thrown away, etc.), then for $ein E$ we only have $color{blue}{chi_{G}^{leqslant}+chi_{Gdownarrow e}^{leqslant}=chi_{Gsetminus e}^{leqslant}+chi_{G/e}^{leqslant}}$, where $Gdownarrow e$ is $G$ with $e$ reversed, $Gsetminus e$ is $G$ with $e$ removed, and $G/e$ is $G$ with $e$ contracted. This does not provide a reduction (even for a simple $G=({u,v},{(u,v)})$, for which $chi_{G}^{leqslant}=chi_{Gdownarrow e}^{leqslant}$ however)...

These are called the weak and strong chromatic polynomial of $G$, respectively; see this paper, for example.

Regarding the deletion-contraction rule, this paper suggests using it to obtain strongly connected components in $G$ by reversing arrows, then contracting those components to a point. I'm not sure how efficient this is when all cycles are long cycles.

Doing this repeatedly will eventually leave us with a graph which is the orientation of a
tree. I don't know what the best thing to do here is. One possibility is to keep flipping edges until all edges are oriented away from a single root vertex. Then we have a recursion for the number of colorings, based on how we color the root and what we do with the subtrees.

Computing $chi^<_G$ is related to counting the topological sorts of an acyclic graph; see this MSE answer. Specifically, a topological sort is a $<$-coloring using exactly $N:=|V(G)|$ colors, which is equal to $chi_G^{<}(N) - N chi_G^{<}(N-1)$. I say this to argue that computing $chi_G^<$ is computationally difficult, because counting the topological sorts is computationally difficult. I expect that something similar holds for $chi_G^{leqslant}$ but don't have a proof.

We can give $chi_G^<(n)$ a recursive formula based on the set $S$ of vertices of degree $0$:
$$
chi_G^{<}(n) = sum_{A subseteq S} chi_{G-A}^{<}(n-1).
$$
This is probably reasonable for evaluating $chi_G^<$ at small values of $n$, but will be obnoxious for actually computing the polynomial. I'm not sure if it's better than brute force in that case.