There are two closely related definitions which satisfy the properties you want.

First, consider the group $\Sigma_k$ of all bijections $\pi: \Bbb Z \to \Bbb Z$ such that $\pi(x+k) = \pi(x)+k$ for all $x$. Note that $S_k$ is a subgroup in $\Sigma_k$ - simply take any permutation of $\{1,\ldots,k\}$ and extend it periodically to all $x$. This group (introduced by Lusztig) is finitely generated and is closely related to affine Lie algebra $\widehat A_k$. The RSK algorithm does not exactly work here, but Lusztig does study the shape of Young diagrams (of what would be resulting two tableaux). The shape is a partition of $k$, and can be described using decreasing subsequences, extending Curtis Greene's theorem (I forgot if this is in Lusztig's paper or my own easy observation).

Second, a somewhat related definition is the group $\Phi_k$ of bijections $\pi: \Bbb N \to \Bbb N$ such that $\pi(x+k) = \pi(x)+k$ for all $x$ large enough. I studied this definition in this paper. This group $\Phi_k$ is also finitely generated. It is very suitable for RSK, which is not always, but sometimes invertible. The asymptotic shape I defined is essentially the same as Lusztig's. Neither I nor anyone else studied the infinite matrix extension. The infinite permutation version is already difficult enough.