I think the issue may be in trying to set the "Name" property using the webservice. I have had some fail doing that.
Given the "Name" is the name of the document, you may have some success with
string targetDocName = "Test1Name.txt";
string destinationUrl = Uri.EscapeDataString("https://someaddress.com/Reports/Temp/" + targetDocName);
string[] destinationUrls = { destinationUrl };
SPCopyWebService.FieldInformation i1 = new SPCopyWebService.FieldInformation { DisplayName = "Title", InternalName = "Title", Type = SPListTransferSpike1.SPCopyWebService.FieldType.Text, Value = "Test1Title" };
SPCopyWebService.FieldInformation[] info = { i1};
SPCopyWebService.CopyResult[] result;
byte[] data = File.ReadAllBytes("C:\\SomePath\\Test1Data.txt");
uint ret = SPCopyNew.CopyIntoItems(destinationUrl, destinationUrls, info, data, out result);
Best Answer
Q. When running with SPSecurity.RunWithElevatedPrivileges (web context) what credentials are being used?
A. The App Pool Identity for the web application running SharePoint.