Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
This is motivated by some representation theory.

The naive idea is to start with the sequence of symmetric functions $s_{n,n}$ and take the Hankel determinants using the inner product (that is product in the group ring of $S(2n)$) instead of the usual outer product. However this doesn't make sense.

Take the $2 times 2$ case. Then the naive determinant is
$$ left|begin{array}{cc} s_{n-1,n-1} & s_{n,n} \ s_{n,n} & s_{n+1,n+1}end{array}right|$$
The inner product of the two diagonal terms is defined but the inner product of the two off-diagonal terms is not.

The idea that I want to test is that this is $sum_lambda s_lambda$ where the sum is over the conjugates of the partitions $4^a2^{n-2a}$.

Any suggestions on how to fix this? If this does get fixed then I would like to know how to calculate the result. The difficulty is that I have not seen an implementation of the inner product in the computer algebra systems I use, Magma and Sage (which I think both use the same source for symmetric functions).