Forget about trying to dye that wheel. Take your transmission shift **** off and take it to a paint store (I used Home Depot) and have them paint match a 3 oz sample of their best latex interior flat paint. Clean the leather and paint it with a small brush. I touched mine up when it had a small area like yours where the leather meets the wood years ago. My touch up is still invisible and the leather looks perfect.

I know that sounds crazy and you will probably want to ignore this suggestion. But, give it a try as you can wipe it off if you find that it doesn't look like you will be pleased. Just remember that the color will change slightly as it drys. With your badly worn wheel your loss will be less than $5 for the paint if you don't like it.