Set NaN to black independent of the LUT

I am working with images that I display using sequentual LUTs, like Rainbow or Cividis. I want to display the background of my images, which I set to NaN value, as black, but I do not want to have black as the minimum value of my LUT.
So, simply put, can I make NaN values black, independent of the LUT?
Thanks a lot,

sample-NaN-to-black.tif (591.7 KB)