The API allows you to set exactly one header, namely Sec-WebSocket-Protocol, i.e. the application specific subprotocol. You could use this header for passing the bearer token. For example:
new WebSocket("ws://www.example.com/socketserver", ["access_token", "3gn11Ft0Me8lkqqW2/5uFQ="]);
The server is expected to accept one of the protocols, so for the example above, you can just validate the token and respond with header Sec-WebSocket-Protocol=access_token.