this is not an answer to the question "what matrix model produces the eigenvalue distribution (2)", but it does explain why (2) cannot be an extrapolation between the $\beta$-Ginibre ensembles.

The eigenvalue distribution in the Ginibre ensemble is remarkably complicated, it only has the simple form (2) for $\beta=2$, see equations 1,2,3 of Eigenvalue statistics of the real Ginibre ensemble. For that reason the "interpolating distribution" (2) is not a natural object for complex eigenvalues --- unlike in the case of real eigenvalues, where it has the same form (1) for $\beta=1,2,4$.

this is not an answer to the question "what matrix model produces the eigenvalue distribution (2)", but it does explain why (2) cannot be an extrapolation between the $\beta$-Ginibre ensembles.

The eigenvalue distribution in the Ginibre ensemble is remarkably complicated, it only has the simple form (2) for $\beta=2$, see equations 1,2,3 of Eigenvalue statistics of the real Ginibre ensemble. If $\beta=4$, instead of a factor $|\lambda_i-\lambda_j|^4$ the repulsion is of the form $|\lambda_i-\lambda_j|^2 |\lambda_i-\bar{\lambda}_j|^2$, and for $\beta=1$ one has three repulsion factors: one factor $|\lambda_i-\lambda_j|^2 |\lambda_i-\bar{\lambda}_j|^2$ between complex eigenvalues $\lambda_i$, one factor $(\mu_i-\mu_j)$ between real eigenvalues $\mu_i$, and one factor $|\lambda_i-\mu_j|^2$ between a real and a complex eigenvalue.

So you see, the "interpolating distribution" (2) is not a natural object for eigenvalue statistics in the complex plane.

Even for $\beta=1$, so for a real matrix, the eigenvalue distribution in the Ginibre ensemble does not have the form (2), so this "interpolating distribution" is not a natural object for complex eigenvalues. For the Ginibre eigenvalue distribution at $\beta=1,2,4$ see equations 1,2,3 of Eigenvalue statistics of the real Ginibre ensemble. As you can see, it has a completely different form for these three values of $\beta$, so there is no natural notion of an "interpolation" --- unlike in the case of real eigenvalues, where it has the same form (1) for $\beta=1,2,4$.

this is not an answer to the question "what matrix model produces the eigenvalue distribution (2)", but it does explain why (2) cannot be an extrapolation between the $\beta$-Ginibre ensembles.

The eigenvalue distribution in the Ginibre ensemble is remarkably complicated, it only has the simple form (2) for $\beta=2$, see equations 1,2,3 of Eigenvalue statistics of the real Ginibre ensemble. For that reason the "interpolating distribution" (2) is not a natural object for complex eigenvalues --- unlike in the case of real eigenvalues, where it has the same form (1) for $\beta=1,2,4$.