Also, what's to say I don't experience the "noise" for longer than a few ms?
@J, it's funny you should ask - I am in the electrical trade.
@J, are you proposing to put a small cap between the pin and ground?
And, I don't think I have the option to software debounce, since sometimes the doorbell doesn't close the relay long enough. And, I can't control how long guests hold the doorbell button. It is a little curious that I get such short contact closures when the doorbell button is tapped quickly, but I'm guessing that has something to do with the chimes' loads... but totally a guess there.