It seems that the categories of log-crystals in the strict etale and Kummer etale topologies are not actually equivalent to one another, contrary to what I expected.
Here's a sketch of a proof. Firstly, we'll restrict attention to the case that $X=S$ is log-smooth, so that the category of log-crystals on $X/S$ in the strict/Kummer etale topology is the same as the category of coherent sheaves in the strict/Kummer etale topology. So it suffices to find an example of an $X$ such that these categories $Coh(X_{et})$ and $Coh(X_{ket})$ of coherent sheaves are not equivalent.
For this, suppose that $2$ is invertible on $X$ and that $t\in\Gamma(X,M_X)$ is an element of the monoid-sheaf on $X$. We define $X_2=X\times_{Spec(\mathbb Z[t])}Spec(\mathbb Z[t^{1/2}])$ (pullback taken in the category of fs log schemes). The projection $\pi\colon X_2\to X$ is Kummer etale, and there is an involution of $X_2$ over $X$ given by $t^{1/2}\mapsto-t^{1/2}$.
Claim: $Coh(X_{ket})$ is equivalent via $\pi^*$ to the category of coherent sheaves $E$ on $X_{2,ket}$ together with an isomorphism $\phi\colon\iota^*E \xrightarrow\sim E$ satisfying $\phi\circ\iota^*\phi = 1_E$.
Proof of claim: The key point is that $X_2\times_XX_2=X_2\times\mu_2$, where the fibre product is taken in fs log schemes, and similarly for the triple fibre product. This then implies that specifying gluing data for a sheaf $E$ on $X_2$ for the map $X_2\to X$ is equivalent to specifying an equivariant $\mu_2$-action on $E$, i.e. an isomorphism $\phi$ as above.
The claim allows one to give plenty of examples of coherent sheaves on $X_{ket}$ which do not arise from $X_{et}$. For instance, suppose that $X=Spec(\mathbb F_p[t])$ is the affine line over $\mathbb F_p$ with divisorial log structure associated to the point $\{0\}$. Let $E_{2,Zar}$ be the coherent sheaf on $X_{2,Zar}$ associated to the $\mathbb F_p[t^{1/2}]$-module $M_2 = t^{1/2}\mathbb F_p[t^{1/2}]$, and let $E_2$ be the pullback of $E_{2,Zar}$ to $X_{2,ket}$. The semilinear automorphism $\psi$ of $M_2$ given by $t^{n/2}\mapsto (-1)^nt^{n/2}$ gives rise to an isomorphism $\phi\colon\iota^* E_2\xrightarrow\sim E_2$ as in the claim, and hence $E_2$ descends to a coherent sheaf $E$ on $X_{ket}$. But $E$ does not arise from a coherent sheaf on $X_{et}$. For, if it did, it would be the sheaf associated to a finitely generated $\mathbb F_p[t]$-module $M$, so we would have $M_2=\mathbb F_p[t^{1/2}]\otimes_{\mathbb F_p[t]}M$ with $\psi$ being the isomorphism $\iota^*\otimes1_M$. But this is not the case e.g. since $M_2$ is not generated as a $\mathbb F_p[t^{1/2}]$-module by its $\psi$-invariant subspace.
Best Answer
A sketch of the proof is as follows:
Consider the site $Y_{syn-cris}$ where the objects are the same as in $Y_{cris}$ but the covering families are surjective syntomic families. Then there are maps of topoi: $\alpha : Sh(Y_{syn-cris})\to Sh(Y_{syn})$ and $\beta : Sh(Y_{syn-cris})\to Sh((Y/W_n)_ {cris})$, defined by $\beta_*(F)(U,T) = F(U, T)$ and $\alpha_{*}(F)(U) = H^0_{syn-cris}(U/W_n,F)$.
Lemma. $R^i\beta_*\mathcal O_{Y/W_n} = R^i\alpha_*\mathcal O_{Y/W_n} = 0$ for $i > 0$.
This implies the result in your question by a standard application of the Leray spectral sequence.
As for the lemma: To prove that $R^i\beta_*\mathcal O_{Y/W_n} = 0$ for $i > 0$, it boils down to checking that if $U$ is an affine in $(Y/W_n)_ {cris}$ then $H^i_{syn}(U,\mathcal O_U) = 0$ for $i > 0$, and this follows the theory of the Cech complex (it is acyclic because of faithful flatness of the syntomic cover).
To prove that $R^i\alpha_*\mathcal O_{Y/W_n} = 0$, we have to check that if $U$ is an open of $Y_{syn}$, and $s\in H^i_{cris}(U/W_n)$ (strictly speaking we have to do the computation with syntomic-cristalline cohomology, but by the previous part, the cohomology groups coincide with the crystalline cohomology groups) then there exists a syntomic cover $U_i\to U$ such that $s\mid U_i = 0\in H^i_{cris}(U_i/W_n)$. Now recall that we can compute this cohomology groups as the hypercohomology groups of the de Rham complex of the divided power envelope of some embedding into a smooth scheme. That means, after shrinking, we can represent $s$ as an $i$-form. We need to find a syntomic cover such that when we restrict $s$ to this cover, it vanishes. To do this, note that $A[T]\to A[T^{p^{-n}}]$ is a syntomic cover that has the property that the image of $dT$ is zero.