hello balky
The Connect Box 3 uses the default DNS-Sunrise-Servers. There’s no way to change this settings on the Connect Box.
As you mentioned, you can disable the DHCP-Server on the Connect Box 3 and use another DHCP-Server on your network.
On Synology DHCP-Server you can set your custom DNS-Server.

The DHCP range can not include 192.168.1.1, this is the default IP of the Connect Box 3 and also the Gateway.

Hope this helps