Insert Cert to Third part

I am working with a partner that requires authentication when we send a request to a web service. We are currently terminating SSL at our BIGIP for our website, then from the web servers we make a call the the partners web service. What I would like to do is attach the cert that sits on the BIGIP to the web request and use it for authentication. I know it is possible if the cert is in the cert store of the IIS server, but would like to attach the one on the BIGIP. Any help would be greatly appreciated.