You should not have to go with thicker strips. When I cut the hole I beveled the edge and that way the strips, once beveled to fit, would just lay in the hole supported by the beveled edge on the cutout. (Does that make sense?)

Looking at your picture, it looks like you glued in an accent strip around the edge. Is that correct? If so, you need to glue another to it before you bevel and bevel that strip so that after glue-up you still have the accent strip showing. All these things are hard to verbalize. Sorry, I don't have a pic to show.