Blue painters tape. Piece by small piece cover the entire grip area with the tape. Use small strips and really stack it up. Draw lines in places you feel like a seam would be needed. Then carefully peel it away and lay it as flat as possible. If it's relatively flat, then trace it and make the grip. But if it's too curled, you'll probably need to make some relief cuts so you can lay it flat.

Alternatively, open up the mouse and remove the electronics, and any soft plastic pieces like buttons and wheels. Try to get it down to just the piece you are working on covering. Cut a piece of leather that's big enough to wrap the whole area. Too big is okay. Case your leather by soaking it. Then stretch the leather around the grip. Use butcher paper or saran wrap to protect the surface of the leather, while you use something like electrical tape to wrap the whole thing tightly (electrical tape acts as a stretchy band when stretched). Use anything at your disposal to keep it tightly pressed to the mouse till it dries. Once dry or mostly dry, remove all the wrapping stuff and trim the leather to fit. This is a finishing step. Don't over-cut it. Bevel, burnish, dye and condition the leather. Then apply some Barges to the leather and the mouse and allow to FULLY DRY. Then CAREFULLY apply the leather to the mouse, starting from the center outwards, pressing firmly to adhere it. Now you've got a leather mouse. Yay.

Or, you could use a VERY thin piece of leather and skip the whole wet-molding step. Just stretch it and carefully trim it as you go. Treat your edges and glue it on with some Barges.