  We consider a family of random normal matrix models whose eigenvalues tend to
occupy lemniscate type droplets as the size of the matrix increases. Under the
insertion of a point charge, we derive the scaling limit at the singular
boundary point, which is expressed in terms of the solution to the model
Painlev\'{e} IV Riemann-Hilbert problem. For this, we introduce a version of
the Christoffel-Darboux identity and combine it with the strong asymptotics of
the associated orthogonal polynomials due to Bertola, Elias Rebelo and Grava.
