What's the current doorbell device like? And are you willing to replace it?
If you're willing to replace it, the easiest way is just to replace it with a non-ringing open/close sensor, and then have that open/close sensor trigger whatever sound you like, and put that on your time schedule.
I have a contact sensor on my front gate that I have smart things monitor and send a text to my phone when the gate opens. The text notification sound is my doorbell.
It would be easy to take any DNLA sound device and have that act as the chime part. Basically just play the sound you specified. So instead of having smart things send a text, smart things turn on the DNLA player.
Aeon been promising a Z wave doorbell with some different sounds for a while now, most recently with a promised delivery date of the summer. If it ever actually arrives, then again, you could easily control the time schedule from smartthings.
so lots of different ways, but all of those require replacing the doorbell button you currently have.
There are some other ways to do it with a relay in between your current doorbell button and your current chime, but that gets more into electrical wiring.