Tracing helped me find what the problem was (Thank you Fabian for that suggestion). I found with further testing that I could get the client certificate to work on another server (Windows Server 2012). I was testing this on my development machine (Window 7) so I could debug this process. So by comparing the trace to an IIS Server that worked and one that did not I was able to pinpoint the relevant lines in the trace log. Here is a portion of a log where the client certificate worked. This is the setup right before the send
System.Net Information: 0 : [17444] InitializeSecurityContext(In-Buffers count=2, Out-Buffer length=0, returned code=CredentialsNeeded).
System.Net Information: 0 : [17444] SecureChannel#54718731 - We have user-provided certificates. The server has not specified any issuers, so try all the certificates.
System.Net Information: 0 : [17444] SecureChannel#54718731 - Selected certificate:
Here is what the trace log looked like on the machine where the client certificate failed.
System.Net Information: 0 : [19616] InitializeSecurityContext(In-Buffers count=2, Out-Buffer length=0, returned code=CredentialsNeeded).
System.Net Information: 0 : [19616] SecureChannel#54718731 - We have user-provided certificates. The server has specified 137 issuer(s). Looking for certificates that match any of the issuers.
System.Net Information: 0 : [19616] SecureChannel#54718731 - Left with 0 client certificates to choose from.
System.Net Information: 0 : [19616] Using the cached credential handle.
Focusing on the line that indicated the server specified 137 issuers I found this Q&A that seemed similar to my issue. The solution for me was not the one marked as an answer since my certificate was in the trusted root. The answer is the one under it where you update the registry. I just added the value to the registry key.
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\SecurityProviders\SCHANNEL
Value name: SendTrustedIssuerList Value type: REG_DWORD Value data: 0 (False)
After adding this value to the registry it started to work on my Windows 7 machine. This appears to be a Windows 7 issue.