HttpWebRequest doesn't seem to be sending a client SSL certificate
You need a certificate with an accessible private key in order to be able to use it as client (or server, for that matter) certificate.
It is OK to store the certificate in the machine certificate store, as long as you grant any user that requires to use the certificate access to its private keys. You cannot authenticate a client using a .cer, because that only contains the public key. You can test everything by installing the client certificate for your user, and accessing the URL using a browser. If you get a popup to select the certificate, it's installed correctly (for that user).