OK, this still doesn't work for longer than a day, so I opened a bug report https://tracker.phpbb.com/browse/PHPBB3-15574
which was promptly closed implying it was a user error rather than a coding problem.
It's also relevant that the "remember me" works fine on desktop browsers (firefox, internet explorer), but not on iphones (and reportedly on some android devices) - the cookie appears to work for some browsers but not others.
Anybody have any insight on how to get this to work?
Edit - the bug has been reopened as an iphone/safari issue, which is the only place I'm able to reproduce the problem myself. I'll report a resolution here if found, whether it's a board configuration or coding issue.