It is unlikely to obtain for such a determinant the sum of all Schur functions indexed by partitions of $2n$ with four parts all even or all odd. Indeed, this sum is already equal to the inner product $s_{n,n}\ast s_{n,n}$ (see arXiv:0809.3459arXiv:0809.3469).
About your second question: you can compute the inner product of symmetric functions in SAGE using the "kronecker_product" command. For instance, compute the inner product of Schur functions $s_{6,2}\ast s_{5,3}$ as follows:
s=SymmetricFunctionAlgebra(QQ,basis='schur')
s([6,2]).kronecker_product(s([5,3]))
You may also use Maple with John Stembridge's package SF and the command "itensor".
