How would I configure a custom action to fire an event that can be picked up in NodeRed?
You could just have Node Red listen for the " home_assistant_start" event as documented here using an events-all node. No need to fire your own event.
Having said that, I use a template sensor that triggers when HA restarts and records the date/time. Then I have NR respond to changes in that value. Then I can also use it to know how recently HA restarted (etc).