There shouldn't be enough dye to trigger a reaction, but other excipients could.
Do you have a problem with corn or potato starch? There may be a disintegrant in the pill too, depending on manufacturer. You need to google the med and the manufacturer, maybe even call them. The starches are made using sulfites, so that's a possibility, but only if you have a bad allergy to it, not just a mild sensitivity.
There could be sucrose, table sugar. And with generics, who knows what's in there and not 100% on the label.
Propylene glycol is another possible thing. It's used in makeup too. Do you have any reactions to makeup?
I would talk with the pharmacist. Also write down the manufacturer and never take that brand again and be leery of anything else they produce.
I would think your doctor should be able to write a different Rx. Definitely call and ask! Insurance might put up a fight, but they can be called by your doctor. You might have to return the leftover pills, probably to the pharmacist. Make sure your pharmacist puts a note in your file about this reaction.
Best wishes and keep in touch!