How can I improve query performance for 200+ million records

I would write your exists query like this:

SELECT stt.sgtin
FROM sgtins_tmp_table stt
WHERE EXISTS (SELECT 1
              FROM sgtins s
              WHERE s.client_id = 4 AND
                    s.sgtin = stt.sgtin
             );

For this query, you want an index on sgtins(sgtin, client_id).