Make sure HttpClient has access to the full client certificate (including the private key).
You are calling GetCert with a file "ClientCertificate.cer" which leads to the assumption that there is no private key contained - should rather be a pfx file within windows. It may be even better to access the certificate from the windows cert store and search it using the fingerprint.
Be careful when copying the fingerprint: There are some non-printable characters when viewing in cert management (copy the string over to notepad++ and check the length of the displayed string).