The only notification the Ring Bridge seems to send are whether it is off line.
Our internet has been intermitemt last fews days so the bridge is sending notifications day (OK) and night (not so good).
My other devices have settings on each device in the app (top right) whether to turn off notifications (for motion etc) - but that option is missing for the Ring Bridge.
None of the other devices send notifications if they are off-line.
Good question, @Ring_In_The_Mountain! As you mentioned, the motion and group toggles can be used to disable most Smart Lighting notifications from the Ring app. The Bridge itself will only send status alerts for online/ offline, as you know. Of course, with the network being intermittent, a constant wifi connection is a requirement of the Bridge for intended experience. Thus, the best first step will be solving connection concerns. As long as a constant network can be maintained with the network and Bridge, it should no longer fall offline or send offline alerts. At this time, there is no toggle or setting that controls this status alerts, but we will certainly share this feedback with our teams! 
I have the same issue, has this been resolved?