Clip a few carpet fibers from a corner of the room and put them into a glass. Pour in a little bit of nail polish remover or acetone to see if the fibers will dissolve. Do this outside and give it some time to work. I used to work in a chemistry lab and we used acetone for dissolving and cleaning all kinds of stuff. Nail polish remover used to be ethyl acetate but I think is now acetone. Just try what you have on hand.
You could also melt some carpet fibers onto a scrap and soak that instead. That would be your next step anyway to see if it would come out of the cotton fiber without affecting the print.