Salt the dye out and adjust the pH when you do the extraction. That always helps push ionic compounds out of water and into solvent!
Then use a HPLC (High Pressure Liquid Chromatograph) machine to separate the compounds.
Only about $10,000 in equipment if you make it youself or buy a used one!
It could be done, if one really wanted too... but I bet you could buy the dyes cheaper by the time one teaches themself how to do it.