All carpets types have different methods of cleaning etc. I would suggest that you contact the carpet store ,perhaps the one the carpet was purchased at if you know and ask them about special cleaners for this.
Although it may be doubtful without professional assistance. You can purchase through them small carpet cleaning kit designed just for your type of carpet. If this is not successful you may need to have that spot cut out by a carpet installer and after removing another replacement section from a hidden area such as a closet re-patch it back in.
I have it too!!! I have tried everything!!!
red is really tough ,but have you tried powdered capture?
Now this is a tough one, did you try some nail polish remover? Not sure if this will work but it ,may remove some amount of red dye from the carpet.
Is the color light enough to bleach? You could always try in a closet or other inconspicuous corner. Similarly, could you cut out a little bit or even patch it with some carpet from the closet or something like that?
I saw this above post before it may help you.
try a product called GOOFOFF. they have 2 types one for painters and one for carpets