S.P.
This is not an"educated response" but just a possibility...
First, call a rug place and ask them what they think
I have been getting together supplies for tie dying and have been learning about the process. With clothes (cotton, which might make a difference) if you want a good dye (not to fade) then you soak in soda ash water 30min. (maybe you could spray it on the area) and that will help the dye adhere to the fibers (increases pH blah blah blah)an dthen use a quality dye (look at an art supply store not crafts) and apply while wet from the soda ash water. If you start with a weaker color than the carpet and get stronger you may be able to patch it...
like I said, I've never done it but if you exhaust all else, what's left to loose...or buy a coffee table.