Jeremy, removing the bezel to access the speedo innards can be brutal... to the bezel and to your wallet. Without a replacement trip odo button thingy on hand, there's no reason to booger up the bezel. Danger, Will Robinson!
One of our forum members, wirespokes, has the skill set to repair these instruments. He may have the push button you desire. I'd bet the button would be part of his repair service charge.
On one of my R65s, I just pushed a black, vinyl vacuum cap into the hole to seal off water. I don't miss an odometer.