Hi Will,
What is the code that you are using to use the $object_id? I've tested this action with a callback function and it displays the object ID as well.
add_action( 'mbct_after_update', function( $object_id, $table, $row ) {
echo "Object ID is: " . $object_id . '<br>';
die( 'Stop here!' ); //for debug
}, 10, 3);
And make sure that you have the latest version of MB Custom Table 2.1.1.