My experience leads me to say that “drilling and filling” is hardly ever successful. There always is a mark where the fill has been made, often there is a colour variation too.
There is another possible solution: turn the whole piece over and take to just below full fuse. This will allow the bubble to begin moving to the “bottom” of the final piece. Then flip over, clean really well and take to a fire polish – quickly up from bubble squeeze temperature – with no soak to give a smooth surface, but not allowing it to get so hot the bubble comes back to the top. This will require observation to stop the temperature rise as soon as the surface is shiny.
As you already know, the prevention of inserting a bubble squeeze, is the best solution.