You can too use glue, just use something flexible which means with silicones in, and do it from the back. Try any silicone glue, and let it dry thoroughly. Shattered furs can sometimes be fixed this way, by gluing to a backing. You can even glue each side of a tear to a separate fabric backing and then after it dries stitch it together as usual. Don’t go overboard on the glue, use as little as possible to get the job done. It’s difficult to get through the glue, use a fine needle and a thimble. Success depends on how far gone the skin really is.
I hand repair all my furs when it comes to re-stitching linings or split seams. A fine #10 or #11 beading needle and either Nymo or Silamide beading thread works well – these are durable yet thin threads.
Furriers tumble furs in a huge drum to clean. The furs are placed in a bag with either sawdust or ground corncobs that has been mixed with solvent, then they are tumbled. The sawdust gets into the fur, the solvent removed dirt. Coarse cornmeal also works well. Not all furriers use a bag, but I would, cuts down on wear and tear on the furs.
Furriers also use a mixture of stuff including silicones to glaze furs. On something you don’t mind experimenting on, you could try spray silicone. You can also try it for softening a stiff skin instead of oil. I haven’t tried either of these yet, just read about them.
Best Wishes,
~ Molly