Did this get resolved because I am having the exact same problem. My Greet worked fine but caused the humming in the chime. I installed the Power Accessory and the hum is gone, but now the chime does not work.
My Greet did come with a Power Accessory so I was having it mailed to me (which never happened). So I resorted to a foam piece between the case and the doorbell assembly. It quieted down the noise to acceptable levels.
Be happy yours works at all. My Greet is just a paperweight.