Can you peer over a tunnel? YES. In fact Core Transit is a tunnel based BGP enabled IP Transit provider of sorts. Is it useful? Maybe / maybe not. It depends on what you're trying to do...
If its cloud registered then, no, which sounds like is the case. The cloud services it needs to make calls / join meetings need internet transport to work. Sounds like you gotta give your network guys a call!
This is likely a firewall issue. I would review this document in detail: https://help.webex.com/en-us/article/WBX000028782/Network-Requirements-for-Webex-ServicesAlso, if you go to Device Settings > Issues and Diagnostics > Webex Connection Details...
The volume buttons are not something you can hide:https://collaborationhelp.cisco.com/en-us/article/nci847eb/Remove-default-buttons-from-the-user-interface-on-Webex-Board,-Desk,-and-RoomSeries-devices
