So I was able to get the MFE to load with the following configuration settings. I didn’t have the {cms|lms}-env tutor patches overridden and that was the issue. As @ghassan mentions here I used the .{{ BASE_DOMAIN}} for the SESSION_COOKIE_DOMAIN value along with the same SESSION_COOKIE_NAME to allow the session to work across both the CMS and LMS applications.
"BASE_DOMAIN": "example.com"
Tutor Plugin Patches
cms-env
lms-env
"SESSION_COOKIE_DOMAIN": ".{{ BASE_DOMAIN }}",
openedx-cms-production-settings
openedx-lms-production-settings
SESSION_COOKIE_DOMAIN = ".{{ BASE_DOMAIN }}"
SESSION_COOKIE_NAME = "sessionid-<named-release>"