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The irreducible representations of full support in the rational Cherednik category ${{𝒪}}_c(W)$ attached to a Coxeter group $W$ are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in ${{𝒪}}_c(W)$ of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup $B_n\times S_k$ in $B_{n+k}$ for $k\ge 2$ and $n\ge 0$.