(0n)$ are of the form $z_0,z_0^{-0},\cdots, z_n,z_n^{-0}$. The corresponding p.d.f. of $z_0,z_0,\dots,z_n$ is proportional to $\Delta^{{\mathrm{HO}}}(\bz;0,0,0)$, and therefore we have $$\left\langle \prod_{i=0}^m \det(I+x_i M) \right\rangle_{SO(0n)} =  \left\langle \prod_{i=0}^m \Psi^{{\mathrm{BC}}}(\bz;x_i) \right\rangle_{n}^{0,0,0} = \prod_{i=0}^m x_i^n \cdot  P_{(n^m)}^{{\mathrm{HO}}}(x_0,\dots,x_m;0,0,0).$$ Here $P_\lambda^{{\mathrm{HO}}}(x_0,\dots,x_m;0,0,0)$ is just the irreducible character of $