I think it's possible to use your FDE key to decrypt the filesystem, autologin to the desktop, unlock GNOME Keyring or a similar keychain application, and use that as a GPG and SSH agent to further encrypt your passwords using pass [1]. Since it's just Git and PGP, there are pass implementations for most OSes. I'm not sure if that counts as "integrating well;" it certainly doesn't work out of the box. I'm using all these components individually, but still typing my password 100 times a day for now.

[1]: https://www.passwordstore.org/