When adding points to a Label Table - Add them first to a Selection Set and then keep adding the points you add to the Table to the same selection Set, then you can use the Selection Set to select the points to run an Export to a CSV file. If you don't like selection sets then you could add the points for the Table to one Layer and those that are not in the Point Table to a Different Layer and use Select by Layer to select the group. You could also use the Point Name (eg Group 1-001) to name the points in the Table differently to those not in the table. You could also use Description 1 or Description 2 to add an attribute to all the Points you place in the Table - like "Table Points" and then you can use Advanced Select to select Points with the Property Description 2 equals Table Points etc to find just those points from all the points in the project.
While the Point Table does have a reference to all the points that it contains, there is no "Select"function to grab those points and add them to the Selection Set
What you can do however is to Select the Table and Right Click to Edit It - all the points listed in the Table are Highlighted graphically, you can leave the Edit command open and now select those points that are highlighted using normal select methods and then create a selection set called Table Points from them - you can always add to that selection set later as needed.
Alan