How to execute a web request on a remote computer such that the cipher suites on the remote computer are used?

Our Windows Server 2008 SP2 (not R2) server does not support SHA-2 (as far as I can tell), therefore authorize.net will no longer accept requests from it as of a few days ago. We have a Windows Server 2008 R2 server that does support SHA-2 and can make the authorize.net requests. I would like to change our WebRequest call so that it makes the request via the remote R2 server.

I tried using a putty tunnel, but it appears that it uses the local cipher suites, etc for making the TLS connection (unless I'm missing something, such as a setting on the proxy tab?).

HttpWebRequest webrequest = (HttpWebRequest)WebRequest.Create(Config.AuthnetUrl);
webrequest.Method = "POST";
webrequest.ContentLength = post_values.Length;
webrequest.ContentType = "application/x-www-form-urlencoded";
StreamWriter myWriter = null;
myWriter = new StreamWriter(webrequest.GetRequestStream());
myWriter.Write(post_values);
myWriter.Close();
String post_response;
HttpWebResponse objResponse = (HttpWebResponse)webrequest.GetResponse();
using (StreamReader responseStream = new StreamReader(objResponse.GetResponseStream()))
{
    post_response = responseStream.ReadToEnd();
    responseStream.Close();
}

How can the web request be made via a remote server, using its TLS handshaking and cipher suites?