To solve this problem, I looked up this question, found @pumpkinthehead’s answer, and realized that all we need to do is find+replace the primary key in each row with the NULL so that mysql will use the default auto_increment value instead.
(your complete mysqldump command) | sed -e "s/([0-9]*,/(NULL,/gi" > my_dump_with_no_primary_keys.sql
Original output:
INSERT INTO `core_config_data` VALUES
(2735,'default',0,'productupdates/configuration/sender_email_identity','general'),
(2736,'default',0,'productupdates/configuration/unsubscribe','1'),
Transformed Output:
INSERT INTO `core_config_data` VALUES
(NULL,'default',0,'productupdates/configuration/sender_email_identity','general'),
(NULL,'default',0,'productupdates/configuration/unsubscribe','1'),
Note: This is still a hack; For example, it will fail if your auto-increment column is not the first column, but solves my problem 99% of the time.