I have been looking into hacking toys recently... Simple ones. I think some of those websites, they could maybe look at your 2012 without the fur and might see the issue. I know one place had a yellow 2012 open with images of it open. I don't know where but you could compare yourself too. Hackers love that sort of thing, fix the problems that are like a puzzle.
There is a older 1998 furby fixer place... Maybe they have updated for the newer versions? They did tests and what not on the broken furbys once they fixed them. I haven't been there in a longtime, to the website. They requested old, broken furby for a collage class.
Not saying send it, but asking questions if they recommend anything.