this depends on the molecular weight and the evaporability of your compounds. I would recommend LC/MS for dye molecules. For GC/MS these molecules should be too heavy and difficult to evaporate.
The separation technique depends on the types of compound present in the Dye sample. HPLC is very frequently used instrument for the separation of high molecular organic compound. Specially for dye molecule, LC/MS is the effective technique.
Assuming the dye is not volatile your next problem will be the relative polarity of the molecules. Some variation of HILIC or Normal phase chromatography will most likely be necessary.
I think and I prefer GPC / MS because the dyes are characterized by their adsorption (UV-VIS) whereas the diférantes function chemical causet diférant fragments by mass spectroscopy as well as a differential retention by gas phase
Indigo carmin dye is a fairly big molecule and a di-sulfonic acid. It would not be easy to make a volatile derivative for GC-MS, but is ready-ionised for LC-MS!
Sample preparation might be a source of difficulty. It may be worth exploring MS and other techniques for examining the surface directly.
On a more general point, spectra obtained by LC-MS don't contain nearly as much structural and identification (fingerprint) as electron ionisation MS. Steroid analysts have the choice: they use fast automated LC-MS for routine assays, but GC-MS for exploratory work and to validate the specificity of the LC-MS methods.