How to keep screen from turning on while unlocked?

When I get a notification from messaging apps like Snapchat, and my device is unlocked but the screen is off, the phone will not buzz but the screen will just turn on. This will happen when I am at a trusted place or I am connected to a trusted bluetooth because I use the smart lock feature. I would prefer to keep using smart lock, but if Snapchat turns the screen on and starts using the camera without me knowing, I'll miss incoming messages and lose massive amounts of battery. **How can I keep using smart-lock AND prevent incoming messages from turning my screen on?** I am on Android 6.0 on a Moto X Pure Edition.