I would hire a professional with a truck mounted rug cleaning system.

I've had success with truck mounted carpet cleaning using "Chem Dry" system. But I've heard many others complain about it. For me, they've done good work. If the stains come back after cleaning, so will the company. They'll come back and re-do the job. Also gave me some spot-cleaner for free. I'm not paid to say this, it's just my experience with a particular company/product/system.I'm not a real stickler. I get my rugs cleaned about once a year.

The stain may be down in the carpet padding and wicking back up after you clean the carpet.

May need to replace carpet padding in the stained areas or put a plastic sheet over the padding to prevent stain from surfacing again. Plastic sheet over the padding will have a slight crinkly sound when walking on the carpet, may bunch up in places over time.