How to transform flat-field corrected spectra to those under Standard illuminant E (ideal equal energy illuminant)
Is the flat field correction the only correction applied to the hyperspectral image? If the camera is not calibrated to radiance values then the fact that you need to apply shift/scale adjustments is normal and expected. The CIE XYZ color of a spectrum of light is defined in terms of radiance, the raw output of a uncalibrated camera is not radiance.

If the camera is radiance calibrated then here are a couple other ideas:

When you compare your computed color vs the known value are you scaling them both so that the CIE Y value of the white patch is 1 (or 100) ?

They need to both be exposure scaled equally to be compared, differences in the absolute scale will affect the CIEDE2000 values.

Also, you say the test target published colors are for illuminant E but was the hyperspectral image acquired under an approximation of illuminant E? If not the colors are not comparable.