The way I do it....

Wax the surface area.

Cut a circle the correct size wanted same as you would for a pop. Align it over the hole (backing still in place) and put three small pieces of blue masking tape at the edges of the circle. Hold it @ 9:00 and put the tape @ Noon, 3, and 6. Verify that if the plastic just brushes against the tape it's centered over the hole. Peel back and fold the backing so that the 3 o'clock side has no backing and you are holding the folded side back on itself. Using care not to stick it down prematurely, align the edges with the tape then tap it down into place. First @ 3, then the other two. Check for wrinkles. Lightly press it into place from 6 to 12. Squeeze out any bubbles with the back of your fingernail. While slowly pulling away the backing, press the mylar down to prevent wrinkles or bubbles. Rub it smooth (even out the adhesive) with the back of your fingernail again.

Use a sharp exacto to trim the plastic from over the hole in the cabinet. Just run the blade perpendicular to the wood. DO NOT lean the knife toward the outside of the hole. You can cut off wood, graphics. Note there will be two or three divots in the edge of the hole in the wood. The flipper button housing has a sharp little protrusion that seats into those divots, but the plastic has made them flat. Cut the plastic the length of that divot. It's a very small cut but important to do to prevent wrinkles.

Now slide the button assembly in and gently spin it until the protrusion seats itself in the divot. While holding it firmly in place reinstall nut.

Worksgoodlastslongtime!

OH, PS: Make the circle big enough to reach out beyond finger length while playing. If players feel that edge they will be tempted to hook a fingernail under there and pick at it.