Removing those teak strips is a simple thing and will make your repair successful. The bungs just split and come out. Do it carefully so you don't damage the edges of the holes. An ice pick is a good persuader. It shouldn't take you 10 minutes to get them out and back out the screws. Replacing the bungs is simple, too. Buy a small bag of the correct size bungs. I use a little thinned
varnish to "glue" the new bung in.
Line up the grain of the bung with the grain in the trim and tap it down tightly with a hard plastic hammer. After it dries you can trim down the new bungs to the shape of the surface of the trim.
I don't hesitate to remove bungs when I need to get behind some trim.
GF