If the 3rd party requires to send a specific client certificate to authenticate, then you are missing a piece of configuration.
First you need to import the provided certificate in your Private Keys, not your Certificates.
Then you need to right-click on the Route via HTTPS assertion and click Select Private Key, select Use custom private key, and then select the key you have imported.
The route will then pass the right private key when sending the request.
See this thread for more info on configuring mutual ssl: