These settings are very aggressive and not something I’d recommend to use in practice. But it should be fine for a test.
K-9 Mail is not using GCM/FCM. If it’s not working with GrapheneOS you’re on your own. We’re not going to investigate why things are not working with a particular custom ROM.
You could enable debug logging and check if there’s anything interesting that could explain the behavior. See https://github.com/k9mail/k-9/wiki/LoggingErrors
I find this to be interesting as I also use K-9 mail with Graphene and I get my notifications just fine. As well as syncing. I only use two accounts but I get them on my lock-screen and notification shade.
Keep in mind that apps AUTOMATICALLY are optimized in Graphene. You have to MANUALLY make K-9 NOT optimized for battery in settings. IF you are using Tracker Control to block trackers, open app up and look for K-9. Click on it and uncheck Essential. This will allow syncing to happen.
The rest is as follows
K9 email go under each individual email profile go to account settings, go to fetching email tab.
Go to Folder poll frequency enable polling every minute for email syncing.
Go under individual email account settings, then to reading email.
Check, allow images
Under each individual email profile go to global settings, then go to privacy tab.
Hide subject in notifications – when device is locked
Hide email client
Hide hostname
Hide special accounts in main app settings
Go to app permissions and deny sensor access
(this allows notification sound when phone is face down)