Install your now drilled Backing Plate over the new Button.
Don't tighten the plate too hard or you will crack the plastic. Use common sense and just snug it down.
Because the new Button's plunger is longer, you may have to shim the Switch Base backwards, so the Switch is not locked on, and you maintain that same Bally spring feel. That is why you want to leave the old switch installed so you have a "feel" to reference.
Back in Bally School, they taught us the flipper switches should make contact in the firs 20% of their travel, the rest of the motion is a "cleaning motion" as the contacts rub together.