That table is for theoretical black body radiator, which you can't buy anywhere. In practice you can't buy a lighting product that actually has a color temperature at all (as expressed in Kelvin). Classic incandescent bulbs come fairly close, though.
White LEDs or fluorescent tubes do not have a color temperature; instead, they come with "color temperature equivalent" also rated in Kelvins, but it only corresponds how the light kind-of-looks like on human eye. Such a table is meaningless; you need to look at the actual spectral curves for the LED you are considering.
All this being said, you can calibrate the light source out of the equation. The poorly defined and overlapping filters on your chosen sensor are a bigger problem because you can't calibrate out the fact that some red light is indicated as "blue".
There is another, simpler way; assuming your sample is not moving, why not use a simple photodiode or similar to just sense light, and use different wavelength LEDs (like red, green, blue, maybe yellow as well) as sources. Switch one on at a time, do a measurement. Spectrum of the LEDs is well defined and there are no extra spikes outside the main peak. This way you can easilly generate like 6-7 different samples along the spectrum, far better than just RGB.