@phill are Fastmail labels supported? I have labels enabled instead of folders but they show as folders in Big Mail.
@dominic are you seeing any Keychain errors after syncing?
If you tap one notification then share logs I can take a look.
@dominic thanks for that.
Your notification extension isn't waking up to download the message in the background (which is why when you sync in the app, then tap, it works).
I would first try restarting your phone to see if it that gets it going again. Failing that you'll need delete the app, then restore back from Big Mail Online.