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 $\beta=1,2,4$.