See Preview | SmartThings-managed Edge Device Drivers
Just follow the “Getting Started” instructions, once you’ve clicked through that you’ll see a list of drivers and you want to select the “Zigbee Button” and install it:
Remove the existing device (button) from ST, the re-add it using the “Scan for nearby devices” option. It should then automatically pick up the edge Zigbee Button when it re-registers it. You can check by going into the device in the ST app and looking at the settings - you should see a “Driver” option. Or use the ST CLI to show devices and it should show ZIGBEE under the Type eg:
