I have the feeling you didn't read my previous mail... ;-)
Between the browser and Janus, you do NOT have SIP over WebSockets. The browser doesn't see any SIP. The only SIP stuff it knows about are SIP URIs, which it uses as identifiers. It uses the Janus API to communicate with Janus and its plugins, and to setup PeerConnections. The SIP plugin, in particular, uses a very simple JSON based protocol to start/stop calls, notify about incoming calls, and things like that. Again, NOT SIP.
Besides, the browser does NOT talk to the SIP server directly, as your diagram seems to suggest on the top part. You should put the SIP server between the gateway and the SIP phone. Think of Janus like a regular SIP endpoint talking "native SIP", on the SIP end.
L.