I like the idea of strips, do you have a hand plane? If you can find a similar color of walnut, and arrange it with similar grain orientation I think you could make an near invisible repair. A 1/4 thick board should be easy enough for you to dial in a hand plane to take a fat shaving of exact thickness. Just go easy with the glue, I have used up to 1/4 water mixed with titebond 2 or 3 to make it spread thinner. You want no real gap and all wood. Id just shove it in with the barrel in place and put some painters tape to hold it until dry. After it's dry trim close but leave a bit extra wood. If you have gaps, rub a bit of glue into them, force it down into the gaps, then wipe the glue off the surface with a damp rag and use a bit of 180 - 240 grit sandpaper. The dust and glue will fill any gaps and make them almost invisible. When it's all cured, sand smooth and finish with a scraper.