The short update:
I was able to move out of the Smart Lighting App and go back to the stock automation rules for lights that need sunrise/sunset time. This means that now it is possible to have local processing rules that use sunrise/sunset.
The long update:
I noticed after the SmartThings Edge rollout that some of my non sunrise/sunset lighting automations were marked as local processing with the little 
icon. So I decided to switch my sunrise/sunset lighting automations back to directly use simple automations and deleted the Smart Lighting App.
After a a few days I noticed that my old issue with my simple automation rules not getting updated sunrise/sunset times. After running into this thread I logged into the Groovy IDE and checked under My location > My home > List SmartApps and noticed I did not have the Weather Station app. To resolve this I added the SmartLighting App to my automations, but did not put any rules in it. This caused the Weather Station app to show up under My location > My home > List SmartApps under the SmartLighting App. This also caused the hub to start tracking sunrise/sunset times correctly, and my plain vanilla rules started getting the correct sunrise/sunset times!
So now I have local processing using simple automation rules for my lighting rules which use sunrise/sunset.