Here's how I think this can be proved based on what Richard Stanley already did in your previous question.

If we take the network in Section 3.1.6, Example 4 part (a) of https://arxiv.org/abs/1409.2562 and remove everything above height $2k+1$, then the entries of your Hankel determinant count the paths from sources to sinks for this network, and hence by the Lindström-Gessel-Viennot lemma, the determinant is the number of nonintersecting families of paths. These nonintersecting families of paths in turn correspond to $k$-fans of $3$-bounded Dyck paths of semilength $n$ (see the explanation/terminology in Ardila). And $k$-fans of $3$-bounded Dyck paths of semilength $n$ are easily seen to be the the same thing as $k$-bounded $P$-partitions where $P$ is the $2n-1$-element zigzag poset. In <a href="https://mathoverflow.net/a/372663/25028">his answer</a> to your previous question, Richard Stanley explained why these $P$-partitions are enumerated by $C^{(2k+1)}_{-n}$.