iframe not reading cookies in Chrome
There is a relatively new cookie attribute called SameSite that was being set by my server automatically. Disabling this (while retaining the settings listed in the question) allows the iframe access to its own cookies in Chrome.
See also Chrome feature status & IETF draft
UPDATE Aug 2020
Chrome now blocks cookies without SameSite set, so you need to explicitly set it to samesite=none and secure=true.