Central temperature of a stellar model with combination of gas and degeneracy
If you have a gas of non-degenerate ions and degenerate electrons, then by definition you have an upper limit to the temperature, since if it were higher the electrons would not be degenerate.

Below this temperature, the electron degeneracy pressure will always greatly exceed any perfect gas pressure from the ions. Since degeneracy pressure does not depend on temperature, then the central temperature could be anything lower than the temperature below which the electrons can be assumed degenerate. For example, the mass, radius and central density of white dwarfs of a given mass will be very similar, although their central temperatures will change by more than an order of magnitude as they cool over billions of years.

If you wish to make the small corrections, either for partial degeneracy (the pressure is then slightly temperature dependent) or to include the ion pressure, then the only way to proceed is add the pressures and then solve the equations of stellar structure numerically.