No tool in Inkscape will get this 100% right without the need to spruce it up afterwards. I would play around with the trace tool until I get a preview result that gets everything right, i.e. the edges - don't worry about the colors because they are easier to fix later - except the gradient in the ear and on the chest. And then you go in afterwards and manually fix/draw over those sections until they look like the bitmap.

I'm not anywhere near my computer so I can't give you more specific instructions.