It appears that in order to get the Live Analysis to show the changes immediately, the fields will need to be populated at the folder level.  The below will populate the TestID field when you click on the Test Set's folder (in order to get to the Live Analysis).  
Try this solution instead:
     Sub CopyTestIDs
            Dim TSFolderID, TestInstID
            'Get the Test Set Folder ID of the current folder'
            TSFolderID =  TestSetFolder_Fields.Field(''CF_ITEM_ID'').Value
            'Get the TestSetTreeManager object to manage test set folders'
            Set tsTree = TDConnection.TestSetTreeManager
            'Get the current folder / node level in the Test Set Tree by the TSFolderID'
            Set tsFolder = tsTree.NodeByID(TSFolderID)
            'Create a list of Test Sets under the current folder / node level'
            Set tsList = tsFolder.FindTestSets('''')
                'Iterate (outer loop) through the Test Sets in the list'
                For Each ts in tsList
                  'Set the TestSetID to the ID of the current Test Set'
                  TestSetID = ts.ID
                  'TestSet (the object for the specific Test Set)'
                   Set TSObj = TDConnection.TestSetFactory.Item(TestSetID)
                  'TSTestFactory (Collection of Test Instanaces objects in the Test Set)'
                   Set  TSTestFact = TSObj.TSTestFactory
                  'List of Test Instances (List of TSTestFactory items)'
                   Set TSTestList = TSTestFact.NewList('''')
                  'Iterate (inner loop) through the TSTestList list of Test Instances'
                   For Each TSTestObj in TSTestList
                       'Populate the user defined field with the Test ID from the Test Instance'
                       TSTestObj.Field(''TC_USER_01'') = TSTestObj.Testid
                       'Post the field change population'
                       TSTestObj.Post
                   'increment the inner loop'
                   Next
                   'Destroy the objects used in the outter loop'
                   Set TSTestList = Nothing
                   Set TSTestFact = Nothing
                   Set TSObj = Nothing
                'increment the outer loop'
                Next
            'Destroy the objects created prior to the iteration / loop'
            Set tsList = Nothing
            Set tsFolder = Nothing
            Set tsTree = Nothing
            'Refresh the display of the module to show the changes in the GUI'
            Actions.Action(''ExecutionGrid.RefreshAllExecGrid'').Execute
    End Sub
Then move the ''Call CopyTestIDs'' from *TestSetTests_MoveTo* to **TestSetFolder_MoveTo** instead.