I'm upgrading my comments to an answer.

As I've mentioned in comments/answers to some of your previous MO questions, this Hankel determinant of Catalan numbers counts fans of nested Dyck paths (see, e.g., Section 3.1.6 of https://arxiv.org/abs/1409.2562), which are the same as plane partitions of staircase shape with bounded, which were first counted by Proctor (see his "Odd symplectic groups" paper https://doi.org/10.1007/BF01404455).

At any rate, this interpretation means that your generating function $D_k(x)$ is the same as the generating function $\sum_{m \geq 0}\Omega_P(m)x^m$ where $\Omega_P(m)$ counts the number of order preserving maps $P\to \{0,1,\ldots,m\}$ for a certain poset $P$ (namely, the staircase partition shape poset; equivalently, the Type A root poset). The general theory of $P$-partitions, as developed by Stanley, thus says that $D_k(x) = \sum_{L} x^{\mathrm{des}(L)}/(1-x)^{\binom{k}{2}-1}$, where the sum is over linear extensions of $P$, when it is naturally labeled (see e.g. Theorem 3.15.8 of EC1). Thus your $A_k(x)$ is the $P$-version of an Eulerian number generating function, and a general result of Brändén (see https://doi.org/10.37236/1866 or https://arxiv.org/abs/1410.6601) says that such poset linear extension descent genearting functions are $\gamma$-nonnegative as long as the poset in question is graded, which it is in this case.

I'm upgrading my comments to an answer.

As I've mentioned in comments/answers to some of your previous MO questions (e.g. Number of bounded Dyck paths with negative length as Hankel determinants and Some nice polynomials related to Hankel determinants), this Hankel determinant of Catalan numbers counts fans of nested Dyck paths (see, e.g., Section 3.1.6 of https://arxiv.org/abs/1409.2562), which are the same as plane partitions of staircase shape with bounded entries, which were first counted by Proctor (see his "Odd symplectic groups" paper https://doi.org/10.1007/BF01404455).

At any rate, this interpretation means that your generating function $D_k(x)$ is the same as the generating function $\sum_{m \geq 0}\Omega_P(m)x^m$ where $\Omega_P(m)$ counts the number of order preserving maps $P\to \{0,1,\ldots,m\}$ for a certain poset $P$ (namely, the staircase partition shape poset; equivalently, the Type A root poset). The general theory of $P$-partitions, as developed by Stanley, thus says that $D_k(x) = \sum_{L} x^{\mathrm{des}(L)}/(1-x)^{\binom{k}{2}-1}$, where the sum is over linear extensions of $P$, when it is naturally labeled (see e.g. Theorem 3.15.8 of EC1). Thus your $A_k(x)$ is the $P$-version of an Eulerian number generating function, and a general result of Brändén (see https://doi.org/10.37236/1866 or https://arxiv.org/abs/1410.6601) says that such poset linear extension descent genearting functions are $\gamma$-nonnegative as long as the poset in question is graded, which it is in this case.