I agree about the bit being the best choice. However, if you have a Dremel tool and the Dremel router base-steady hands help too- you can do a nice job. I used this method to install some brass inlays in a ukulele and a guitar I recently built. I do not know what kind of grip material you have, I assume wood, but here's a couple of tips that I picked up that helped me.
Make your cuts a little at a time, being cognizant of the correct direction to feed the router into the work. Check your fit often, and make the diameter a bit shy of the diameter needed. You can finish it off by hand with either a scraper or sandpaper wrapped around a dowel. Since you want to install something-coin?-and keep the face then you have to get the depth just right so that any sanding you do won't erase the face of the inlay.
If you happen to get the diameter a little too big, you can fill the gap with sanding dust mixed with your choice of glue. I've used Tite-Bond, and thin super glue. Once the dust-glue mixture is dry, you can carefully sand it and the filler will be invisible. One other thing that helps,especially if you have a snug fit, is to taper the edge of the inlay and the cut a slight amount. This will help seat the inlay without mashing the cut.
This probably sounds like a lot of work, but it really isn't too bad. I would practice on some scrap material first.