The one Finger breadth to fit between the space is an easy measurement you'll find at TW and several references. It works well.
If you are pressed to get 4 1/4, it is 1/8 inch bigger than 4 1/8. You can rip the 4 1/4 original grip off and use a thinner grip to wrap the handle. If you can wait, then wait from the manufact. for 4 1/8. Have you checked several vendors yet?