How can I remove a coffee stain from carpet?
If the pros couldn't do much, It sounds like the stain is there for good. Some suggestions:


small throw rug/welcome mat. Just cover it up!
bring in a pro to swap that bit of carpet with maybe something in a closet, or just a far corner.  A good carpet pro should be able to seam it all back together. 
dye the entire carpet a shade or two darker. I know such services exist, though I have not used them and can't vouch for them.