Finally figured out a work around. If anything in this should be accomplished differently, please correct me.
public static void HandleSaveDialog(ArtOfTest.WebAii.Controls.Xaml.Wpf.TextBlock element, string filePath)
{
// Create an instance of the Manager class to allow access to its functions.
Settings mySettings = new Settings();
Manager myManager = new Manager(mySettings);
// Connect the manager to the first running instance of dTIMS.
myManager.ConnectToApplication(System.Diagnostics.Process.GetCurrentProcess());
// Create the dialog handlers.
GenericDialog genDialog = new GenericDialog(myManager.ActiveApplication, "Save As", false, 6);
SaveAsDialog saveDlg = SaveAsDialog.CreateSaveAsDialog(myManager.ActiveApplication, DialogButton.SAVE, filePath);
// Add the save dialog handler to the monitor, and start the monitor.
myManager.DialogMonitor.AddDialog(saveDlg);
myManager.DialogMonitor.Start();
// Click on the element passed in that will create the dialog.
element.User.Click();
// Wait until the dialog is handled, and then stop the monitor.
saveDlg.WaitUntilHandled(30000);
myManager.DialogMonitor.Stop();
// Add the generic dialog handler to the monitor, and start the monitor.
myManager.DialogMonitor.AddDialog(genDialog);
myManager.DialogMonitor.Start();
// Sleep for 3 seconds to give an overwite dialog to occur.
System.Threading.Thread.Sleep(3000);
// Stop the monitor, and remove both dialog from it.
myManager.DialogMonitor.Stop();
myManager.DialogMonitor.RemoveDialogs(saveDlg, genDialog);
}