As I guessed in a comment above, there is apparently a symplectic/orthogonal hook-content formula which uses these notions. See "Hook-content Formulae for Symplectic and Orthogonal Tableaux" by Campbell and Stokke https://doi.org/10.4153/CMB-2011-105-7.

Warning: they seem to have withdrawn the arXiv version of their paper (https://arxiv.org/abs/0710.4155) for reasons that are not clear to me.

As I guessed in a comment above, there is apparently a symplectic/orthogonal hook-content formula which uses these notions. See "Hook-content Formulae for Symplectic and Orthogonal Tableaux" by Campbell and Stokke https://doi.org/10.4153/CMB-2011-105-7.

Warning: they seem to have withdrawn the arXiv version of their paper (https://arxiv.org/abs/0710.4155) for reasons that are not clear to me.

EDIT: Regarding priority, I see Campbell and Stokke cite El Samra and King for the formula for the evaluation of the symplectic/orthogonal characters at $(1,1,\ldots,1)$ (i.e., pure counting of tableaux); what they do that is new is the principal evaluation $(1,q,\ldots,q^{n-1})$ (i.e., $q$-counting). Perhaps the arXiv withdrawal is about what Per said: copyright.