I've looked on this German archive site to see if your relatives are mentioned - I looked for "Hoppe" as the other surname brought up far too many results. Although that surname is included, unfortunately not the girl you're looking for.
Although you state your ancestors came from Hanover I think maybe they lived in the Kingdom of Hanover, which we now know as Saxony and Lower Saxony in Germany. The royal family would have the use of more than one castle which will make your search a bit more difficult. This webpage shows some Saxony castles(schloss)in Germanyhttp://www.germany.co.uk/Culture/Castles/article,Charlottenb...
and this page shows that the king was born in a Berlin hotel altho I don't know why as there's a castle in Berlin http://www.answers.com/topic/george-v-of-hanover