Lightstreamer does not perform client authentication on its https connections,
nor does it forward the client X.509 certificate to the Metadata Adapter upon session initiation requests; the latter might be a feasible extension.

Request to the client for HTTP authentication is also not supported.
If your client can manage to force the inclusion of the certificate in the request header of all the http or https requests performed by the client library
(like a cookie, but, preferably, not as a cookie),
then your Metadata Adapter will receive it (only upon session initiation requests) in notifyUser, beside username and password, through the httpHeaders parameter.

Otherwise, the certificate would have to be embedded in some way into the username or password field.