Query with two posttypes and two different meta_keys

Have you tried this?

 'meta_query' => array(
        'relation' => 'AND',
        array(
        'relation' => 'OR',
        array(
            'key' => 'start_date', /// or 'end_date' for cpt 'group_of_events??
            'compare' => 'BETWEEN',
            'type' => 'DATE',
            'value' => array($filter_start_date, $filter_end_date),
            ),
        array(
            'key' => 'end_date', /// or 'end_date' for cpt 'group_of_events??
            'compare' => 'BETWEEN',
            'type' => 'DATE',
            'value' => array($filter_start_date, $filter_end_date),
            ),
        ),
        array(
            'key' => 'type',
            'compare' => 'IN',
            'value' => $filter_type,
            ), 
        ),

then you can change the start_date of the events to another key.

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