I found advice online about removing commercial-brand Silly Putty, which involves freezing the putty off and then using WD-40 to remove the stain. I'm not sure what's in Silly Putty. The troublesome ingredients in homemade putty are white glue and food coloring. I wasn't too worried about the glue, because it just peels off when it's dry but I was nervous about the food coloring.
I've seen experts lift a wine stain off carpet with salt. I thought I would give it a try. No biggie if it didn't work, as we could just dust Kate off. After much picking and scraping at Kate's fur, it came off. Kate is twelve years old though and extremely well-loved so I had to pick gingerly at her threadbare fur. Also, she has a hole on her chest so I had to be careful about dropping salt into that. You may be able to go faster with a newer stuffed animal. Be patient: it took me three passes: once with course salt, once with fine salt, finally with anti-bacterial soap.
|Phase 1: course salt.|
|Don't look Chloe!|
|Eat salt, Kate!|
|Phase 2: fine salt.|
|Phase 3: foamy anti-bacterial soap.|
|Whew! Crisis averted.|