I have a concealed cistern behind MDF, which has been tiled.

I am fitting a new flush valve in place of the old syphon. It's mostly
done, but I need to recess the button. That needs a 40mm hole through
the tiles and MDF, in place of the existing rather ragged 30-35mm hole.

What's the best way of doing that? 40mm drill will tend to wander. I
have a 1/2" belt sander, but that'll take ages. Any suggestions please?