Once I had grease on a couch and the company came to clean it...the guy noticed a rust spot on my carpet and used the same spray on it as he did the couch.
Of course I don't know what that spray was but the rust definitely came mostly all the way out.
It was a aerosol of some kind. It took the grease out no problem (it was oil or something from the delivery truck) and I think the rust might have come all the way up had the guy been willing to work at it. I didn't press him since that wasn't his job at all. He had just tried it on the rust since I asked what it was and if it would take up the rust stain.
So basically long post with no answers.
Maybe though MeanLaureen's DH will know of an aerosol that will take up grease that might work on rust???