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 $C^{(2k+1)}_{-n}$.

**EDIT**:

For clarity, here's an example of the kind of network + families of nonintersecting paths:

[![enter image description here][1]][1]

This depicts the things counted by $C^{(7)}_{-4}$. We convert the nonintersecting lattice paths to the sequences mentioned in Richard Stanley's answer by stacking the $k$ orange $3$-bounded Dyck paths on top of one another (they are a fan, i.e., nest, by the nonintersecting condition), and then reading off $1$ plus the number of Dyck paths below the "circles" (which form a length $2n-1$ zigzag poset). In the depicted case we have $(a_1,\ldots,a_7)=(3,4,1,1,1,2,1)$.

This raises an interesting possibility:

Let's let $\mathcal{D}_k$ denote the infinite network where we take a diagonal slice of width $2k+1$ of the 2D grid, with all edges directed right and up. The above discussion explains that there is a relationship (in fact, a "reciprocity" relationship) between counting families of nonintersecting paths in this network with $1$ source and $1$ sink (these are what $C^{(2k+1)}_{n}$ count), and counting such families with $k$ sources and $k$ sinks (these are what $C^{(2k+1)}_{-n}$ count).

**Question**: Is there a similar "reciprocity" relationship between counting families of nonintersecting lattice paths in $\mathcal{D}_k$:
 - when we have $i$ consecutive sources, then a gap of some size, then $i$ consecutive sinks;
 - and when we have $k+1-i$ consecutive sources, then a gap of some size, then $k+1-i$ consecutive sinks?

Note that when we have $k+1$ consecutive sources and then a gap and then $k+1$ consecutive sinks, there's a unique family of nonintersecting lattice paths in $\mathcal{D}_k$; this "agrees" with the fact that there is a unique such family when we have 0 sources and sinks as well. In other words, we can say **yes** to this question when $i=0,1$.


  [1]: https://i.sstatic.net/oUKJ3.png