My experience with food dye and carpet isn't great, but I'll tell you what I know.
When I asked, Rouge suggested trying club soda and blotting with a sanitary pad as it absorbs the moisture and hopefully remove the stain. But We had red food dye and it is the hardest colour to get out. I had a professional come in and try to get the stain out, but ultimately we had to replace the patch of carpet where she spilt it.