Window function correction when using frequency domain likelihood

I was wondering if someone could tell me or point to resources which discuss what kind of of windowing corrections factors one should use when using a frequency domain Whittle likelihood for spectral estimation (or frequency domain parameter estimation in general).

I know that when using FFTs to estimate PSD, the square of the absolute value of FFTs is scaled by $$\frac{2}{w^2}\frac{1}{f_s T}$$, where w is the window function. Should the same scaling factor be used in the Whittle likelihood too? I tried this scaling with simple spectral models with a single parameter and even though the spectra match what I expect, the likelihood estimates for the spectral parameter do not match what I expect.

The "data" is colored gaussian data simulated using the correct spectral values. I have tried a handful of simple models, and I never had good consistent recoveries. I have been trying to figure this out for a few weeks with little progress and lot of frustration. Any help will be greatly appreciated. Thank you!