I think a plug would look nicer than a dowel. If you?re cutting them with a plug cutter, you can use any wood you want. If you go that route, make sure you align the grain in the plug with the grain in the surrounding wood for that extra touch. You can make a guide and use a hand drill, but cutting plugs with the plug cutter will be easiest with a drill press.
A tapered plug usually makes for a tighter fit, but if your hole is only 1/8? deep, the taper might not matter. For what it?s worth, I?d try to go a little deeper even if it?s only 3/16?. Just a little extra surface area for the glue so the plugs don?t pop out with seasonal changes.