OK, so the Next method isn't just for slides. It's also for animations. Here's the code that works:

      Application.Activate
      ActiveWindow.Panes(1).Activate
      Windows(1).View.GoToSlide slideNum
      ActivePresentation.SlideShowSettings.Run
      SlideShowWindows(1).View.GoToSlide slideNum
      SlideShowWindows(1).View.Next