I don't know an algorithm, but here's a possible approach. As Richard and Lee have observed, one may assume that $\psi$ is pseudo-Anosov. In that case, the mapping torus $T_\psi$ is a hyperbolic 3-manifold fibering over $S^1$, with fiber $S$. There is a short exact sequence $\pi_1(S)\to\pi_1(T_\psi)\to \mathbb{Z}$.

Here's a characterization of the translation length on the curve complex in terms of the topology of $T_\psi$. The fiber $S$ represents a homology class $[S]\in H_2(T_\psi)$. Let $\Sigma \looparrowright T_\psi$ be an immersed connected surface, such that $[\Sigma]=k[S]$ and such that $\chi(\Sigma)=k\chi(S)$. Moreover, assume that the composite map $\pi_1(\Sigma)\to \pi_1(T_\psi) \to \mathbb{Z}$ is non-trivial, so that $\Sigma$ is not homotopic to a finite-sheeted cover of $S$. Let $K(\psi)$ be the minimal such $k$, and let $D(\psi)$ be the minimal such $k$ so that the surface has only double curves of intersection. Clearly $K(\psi)\leq D(\psi)$.

Claim: The curve complex translation distance of $\psi$ is $=D(\psi)$.

One direction: Let $k$ be the translation length of $\psi$. There exists a sequence of non-separating curves $c_1,c_2,\ldots, c_k \subset S$, such that $\psi(c_1)=c_k$, and $c_i \cap c_{i+1}=\emptyset$. One creates a surface $\Sigma\subset T_\psi$ by taking $k$ copies of $S$, $S_1 \sqcup \cdots \sqcup S_k \subset T_\psi$ in circular order. Cut out annular neighborhoods of $c_i, c_{i+1}$ inside $S_i$, and insert cross annuli between $S_{i-1}$ and $S_i$ (taking indices $\mod k$) between the 4 copies of $c_i$. This construction generalizes a construction of Cooper-Long-Reid. One can see that the resulting surface has the properties above.

Conversely, if one has such an immersed surface with only double curves, one may cut and paste the self-intersection curves to get $k$ parallel copies of $S$. The cross cut curves gives a sequence of closed curves in $S$, which one can prove using the homology condition forms a closed loop in the curve complex $\mod \psi$.

I don't know yet how to make this criterion into an algorithm. I think there is an algorithm to compute $K(\psi)$. For a given genus $g$, Canary proved that there are only finitely many homotopy classes of immersed surfaces of genus $g$. I think this proof can be made effective, and should give one a method to compute $K(\psi)$. This would at least give an algorithmic lower bound, since $K(\psi)\leq D(\psi)$. Also, there is a constant $0< c_S <1$ such that $D(\psi)\leq c_S K(\psi)$ (this may be proved using hyperbolic geometry techniques).

One could try to algorithmically to construct all surfaces realizing $K(\psi)$, and then try to homotope them to have only double curves of intersection, e.g. using normal surfaces. However, there is a result of Gulliver-Scott that an immersed surface with only double curves of intersection might have a minimal area representative which has triple points. So I don't know yet how to make an algorithm by computing $D(\psi)$ using this approach.

I don't know an algorithm, but here's a possible approach. As Richard and Lee have observed, one may assume that $\psi$ is pseudo-Anosov. In that case, the mapping torus $T_\psi$ is a hyperbolic 3-manifold fibering over $S^1$, with fiber $S$. There is a short exact sequence $\pi_1(S)\to\pi_1(T_\psi)\to \mathbb{Z}$.

Here's a characterization of the translation length on the curve complex in terms of the topology of $T_\psi$. The fiber $S$ represents a homology class $[S]\in H_2(T_\psi)$. Let $\Sigma \looparrowright T_\psi$ be an immersed connected surface, such that $[\Sigma]=k[S]$ and such that $\chi(\Sigma)=k\chi(S)$. Moreover, assume that the composite map $\pi_1(\Sigma)\to \pi_1(T_\psi) \to \mathbb{Z}$ is non-trivial, so that $\Sigma$ is not homotopic to a finite-sheeted cover of $S$. Let $K(\psi)$ be the minimal such $k$, and let $D(\psi)$ be the minimal such $k$ so that the surface has only double curves of intersection. Clearly $K(\psi)\leq D(\psi)$.

Claim: The curve complex translation distance of $\psi$ is $=D(\psi)$.

One direction: Let $k$ be the translation length of $\psi$. There exists a sequence of non-separating curves $c_1,c_2,\ldots, c_k \subset S$, such that $\psi(c_1)=c_k$, and $c_i \cap c_{i+1}=\emptyset$. One creates a surface $\Sigma\subset T_\psi$ by taking $k$ copies of $S$, $S_1 \sqcup \cdots \sqcup S_k \subset T_\psi$ in circular order. Cut out annular neighborhoods of $c_i, c_{i+1}$ inside $S_i$, and insert cross annuli between $S_{i-1}$ and $S_i$ (taking indices $\mod k$) between the 4 copies of $c_i$. This construction generalizes a construction of Cooper-Long-Reid. One can see that the resulting surface has the properties above.

Conversely, if one has such an immersed surface with only double curves, one may cut and paste the self-intersection curves to get $k$ parallel copies of $S$. The cross cut curves gives a sequence of closed curves in $S$, which one can prove using the homology condition forms a closed loop in the curve complex $\mod \psi$.

I don't know yet how to make this criterion into an algorithm. I think there is an algorithm to compute $K(\psi)$. For a given genus $g$, Canary proved that there are only finitely many homotopy classes of immersed surfaces of genus $g$. I think this proof can be made effective, and should give one a method to compute $K(\psi)$. This would at least give an algorithmic lower bound, since $K(\psi)\leq D(\psi)$. Also, there is a constant $0< c_S <1$ such that $D(\psi)\leq c_S K(\psi)$ (this may be proved using hyperbolic geometry techniques).

One could try to algorithmically to construct all surfaces realizing $K(\psi)$, and then try to homotope them to have only double curves of intersection, e.g. using normal surfaces. However, there is a result of Gulliver-Scott that an immersed surface with only double curves of intersection might have a minimal area representative which has triple points. So I don't know yet how to make an algorithm by computing $D(\psi)$ using this approach.