Cat urine is oily, believe it or not, which is why it is so difficult to remove and why the Ammonia worked for Double C. The trouble I see is soaking the rawhide well enough to float out all the urine is likely going to change the texture of the rawhide. Who knows. since natives used urine turned to ammonia to make white leather, it might even tan it. I suppose if you've thrown the thing away a few times it can't hurt to try.
Sudsy Ammonia (found at the dollar store)
then
Vinegar and Baking Soda. (place vinegar in container and soak, then rinse, shake dry... then bury in a box full of baking soda for a few days)
Finally..
A good dousing of Listerine (the old fashioned flavor, if you can find it)
I mean for you to use all three to REALLY get this thing clean.