= Solution
The off-diagonal auxiliary integrations impose $M^i{}_j=0$ for $i\ne j$, by the Fourier representation of a <Dirac delta function>. After this constraint, $M=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$ and
$$
[M,C]^i{}_j=(\lambda_i-\lambda_j)C^i{}_j,\qquad
\operatorname{tr}(B[M,C])=\sum_{i\ne j}B^j{}_i(\lambda_i-\lambda_j)C^i{}_j.
$$
The <Grassmann Gaussian integral> over each off-diagonal pair gives its coefficient. Consequently the ghost determinant is
$$
\det{}'\operatorname{ad}_M=\prod_{i\ne j}(\lambda_i-\lambda_j)
=(-1)^{N(N-1)/2}\prod_{i<j}(\lambda_i-\lambda_j)^2.
$$
The prime omits the diagonal directions, which were excluded from the outset. The constant sign depends on the <Berezin integral> ordering and can be absorbed in normalization. Thus the <Vandermonde determinant> squared is the eigenvalue measure factor in this <matrix diagonalization ghost determinant>.
Up to an eigenvalue-independent constant, the remaining integral is $\int\prod_i d\lambda_i\,e^{-S_{\rm eig}}$, where
$$
\boxed{S_{\rm eig}=N\sum_i\left(\frac{\lambda_i^2}{2}+\frac g4\lambda_i^4\right)
-2\sum_{i<j}\log|\lambda_i-\lambda_j|.}
$$
\b[The determinant supplies logarithmic eigenvalue repulsion]. An ordering restriction on the <eigenvalues> changes only a constant factorial; no remaining eigenvalue integral needs to be performed. The off-diagonal bosonic contours and the displayed normalization are understood in the usual Fourier-delta prescription.
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