Do you just want a proof that some bijection exists, or do you want a proof that the RSK bijection itself is a bijection? If the former, this is equivalent to the Cauchy identity in the theory of symmetric functions, for which many proofs have been given. For instance, it is equivalent to decomposing the symmetric algebra of $V\otimes W$ (where $V$ and $W$ are finite-dimensional complex vector spaces) into irreducible representations of GL$(V)\times$ GL$(W)$, for which a direct argument is possible.

Do you just want a proof that some bijection exists, or do you want a proof that the RSK bijection itself is a bijection? If the former, this is equivalent to the Cauchy identity in the theory of symmetric functions, for which many proofs have been given. For instance, it is equivalent to decomposing the symmetric algebra of $V\otimes W$ (where $V$ and $W$ are finite-dimensional complex vector spaces) into irreducible representations of $\operatorname{GL}(V)\times \operatorname{GL}(W)$, for which a direct argument is possible.