There is an existing sandcastle example that shows how to access the value of custom properties. In the example, this code is in the event handlers for the toolbar buttons. In short, custom properties are accessed exactly like any other entity property - by calling getValue to compute the current value for the given time.
I found out about an hour after I wrote that post and deleted it. Sorry it made it to the summary anyway.
My main problem was I could not use a lookup by ID as I need to get the properties attached to a clicked object. Hence my question whether you could attach custom properties to the document instead of a specific entity. Have solved it by inserting a dummy entity at the beginning of my CZML now.