I have a custom container offering bowl to Nocturnal. I want, once I exit from the container add/remove item screen, to use RemoveAllItems to empty it. Ideally later I'll try to add a blessing for giving items, but for now I'm not sure what Event I would use to put in the script. OnActivate doesn't seem right since I want it to happen after I close the screen and OnClose seems to be refering to doors not containers.

Try OnItemAdded, possibly with an event filter. http://www.creationkit.com/OnItemAdded_-_ObjectReference

If needed, you could add a Utility.Wait(0.01) or something, which will pause the script until the menu closes.