@PerAlexandersson It may be easier than that - since the second Schur function in each term is a column Schur function, one can use Pieri's rule. Let $\lambda=(\lambda_1,\lambda_2)$ be a partition with two columns, and let $\mu=(\lambda_1+1,\lambda_2)$. One needs to construct an injective map from the set of partitions obtained by adding a vertical strip of length $k$ to $\mu$ into the set of partitions obtained by adding a vertical strip of length $k+1$ to $\lambda$.

@PerAlexandersson oeis.org/A054225.

Thanks. The strategy works. The case where $k$ and $l$ are both odd is resolved by looking at $p_3(k,l)$.

@BrianHopkins Thanks. Corrected.

@LSpice This approach is fleshed out in my book (imsc.res.in/~amri/rtcv) in the $q=1$ case to recover many classical results about representations of $S_n$, such as the classification of irreducibles, Young's rule, and twisting by the sign character. You'll also find a beautiful exposition in Howe and Moy to representations induced from tensor powers of a single cuspidal representation.