Thanks for that info and choosing the Apple Support Communities. If I understand correctly, you are unable to use the 3D Touch feature with Mail notifications on your iPhone, even after trying some good troubleshooting steps. If you haven't already, please restart your device as iOS clears cached data and does other system optimizations when you turn your device off and on.
Restart your iPhone, iPad, or iPod touch
Force an app to close on your iPhone, iPad, or iPod touch
If you're still having this issue, please use iTunes on your Mac or PC to backup and restore your iOS software to factory settings, as this process erases the information and settings on your device and installs a fresh copy of the latest iOS software. Temporarily setup your phone as new to test your issues, and then restore your backup and test the issues again.
How to back up your iPhone, iPad, and iPod touch
Import photos and videos from your iPhone, iPad, or iPod touch
Restore your iPhone, iPad or iPod touch to factory settings
Restore your iPhone, iPad, or iPod touch from a backup