Unless something is very wrong with the camera design, the noise you see
with a well illuminated sensor should be massively dominated by the
Poisson distribution of the signal itself

This is only the case in the absense of fixed pattern noise.

Fixed pattern "noise" isn't noise, however.

That depends on how you define noise. However, the OP is interested in
calibrating his detector, and he won't find he has a poission
distribution untill after he makes the correction for fixed pattern
noise.

You can trivially remove it
by applying dark frames, and then you're left with actual noise.

Fixed pattern noise (at least in the way I use the term) is removed
via flats, not darks.