Nevermind...I figured it out. Someone else might benefit though...
Blank parameters cannot be omitted from the query string. The two blank parameter boxes (Product Desc and Product ID) must be passed in with no value in order to make the render command work. It appears this is necessary even though those parameters are not required to run the report.
So the working query string has two (blank) parameters added on the end and looks like this:
http://reportserver/ReportServer/Pages/ReportViewer.aspx?%2fSales+Mgr+Reports%2fCustomer+Product+Variance+with+Net+Margin+Report&rs:Command=Render&rs:Format=HTML4.0&ReportType=PD&SortType=P&CurBeginDate=04/15/2013&CurEndDate=04/15/2014&CusCode=XXXXXXXXXX&ProdDesc=&ProdID=