I use a wood filler putty that dries rather than sets to hard. It is soft enough to remove with your fingernail when "hard." I put a blob on the grip roughly where I think it should be thicker, put on a vinyl glove, grip the gun to shape the putty, remove my hand hand (it may help to spray the glove with some wax or WD40 but not a big deal), reshape with a finger to fix any areas that got messed up removing the hand and/or to smooth, let dry, and then use file, fingernail, or whatever to smooth and shape.
If you got it too thick, it is easy to remove. If too thin, just do it again.
To finish, use some stain and penetrating finish.