Thanks Moose! This is one of the reasons why I posted the diagram. 
That makes sense that the capacitors would keep power to the shield in the configuration I have drawn. I would also like to know if dropping the 12v to the SSR will cut power to the router, we’ll have to test that out. When you say it would take another relay to shut the router down, would it simply be another SSR?
As far as hardware, are you refering to one of these? I’m still a bit new to all the different electronic components.