Hope you are calling the third party web service from your server end, your integration should look like;
[Browser] ---[your server]---when token not present---[third party security toke service]
[Browser] ---[your server]---when token present---[third party service]
your server should use a http-client to call the third party service, where you should add the certificate when calling the security token service. This should return a token which should be set in your browser cookies.
WebRequestHandler handler = new WebRequestHandler();
X509Certificate2 certificate = GetMyX509Certificate();
HttpClient client = new HttpClient(handler);