Such a rescaling is done for the symplectic Schur functions by Bisi and Zygouras in https://arxiv.org/abs/1703.07337, in e.g. theorem 4.2 (following O’Connell-Seppalainen-Zygouras for the classical Schur case). The functions are called $ s^{\mathrm{cont}}_\lambda(x) $ for “continuum” version of the Schur functions. The analogue of the HCIZ integral (in the GT form) are given, for Whittaker functions, by identities of Stade and Ishii. Such a rescaling is called there “$0$-temperature limit, in the context of the log-Gamma polymer, so, no, this is not really the limit performed to go from discrete to continuous statistical mechanics models, more to stay on the same model but replace the classical $ (+, \cdot) $ algebra by the max+ algebra (the tropical version of the statistical model). One stays on the same discrete grid, but the log-gamma partition function becomes the last passage percolation time (see proposition 4.1).
For a nice review on these continuous Schur functions and their links with stochastic integrable models, see the survey by O’Connell https://arxiv.org/pdf/1201.4849.pdf (it is named $ J_\lambda(x) $ here).

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