There exists a rust remover product at carpet cleaning supply stores like jon-don or cobbs that I use. Not sure if they will sell to just anybody though. I had to set up an account with my company name.
I had some rust stains on my tan carpet on back deck and I grabbed the "Shout" out of the laundry room. It took them right out. Sprayed them pretty good and scrubbed it in then after stains left I diluted the carpet with water to get the shout back out. Never know they were there. On the bottle it says to try in a non visible area on clothes first but I just went after it thinking it couldn't look any worse than rust on light carpet. Worked great.