I managed to hack something together to achieve this. It's not elegant, but may point out an avenue for further exploration.
I ended up manually data-binding my RadMenu ("SecondLevelMenu") in the Page_Load event, something like this:
|SecondLevelMenu.DataSource = this.RootMenu.GetSecondLevelMenuItems(Request.Path);
The details of RootMenu don't really matter; it's just an object that contains its own internal model of my hierarchical menu structure, (a sort of Site map). The method "GetSecondLevelMenuItems" takes the given path, works out where we are in the hierarchy and and returns an array of second-level objects which are appropriate for the current page. It actually returns an array of SiteDataItem objects copied from the Telerik article "Binding to object-based datasources
The main trick was that any SiteDataItem objects returned by my GetSecondLevelMenuItems() method had to have the ParentID set to null, in order that they displayed as if they were first-level menu items.