OK, here is the answer :)
Spectral data from most spectrophotometers is already corrected in so far that the hardware illuminant and angle dont matter.
What you do is just use the observer functions for every single angle/illuminant, as written in ASTM E308, to convert the spectral data to XYZ instead of only using the table which corresponds to the hardware illuminant/angle.
Thats a lot of reference values but it works perfect.