Last update from CW Support:

Service Ticket#13409638/Mange New Chat Notification

Hello Milosz,

I am just reaching out to you regarding your issue with chat. I was doing research into this issue and unfortunanlty the chat notification is a hard-coded sound and you are not able to update/change it either in the system or the back-end. I can see where this feature would be extremely useful. Please consider creating or up-voting an existing Enhancement at ConnectWise.com/Enhancements.

No traction on this at all. This alone makes us want to switch to chatgenie as it fully integrates with MS Teams and you're also able to updates tickets on the fly etc... 

https://chatgenie.io/