Would the White House be an example of a compound proper noun?
I would argue no.
I would say it is simply a proper noun, as it is unique. A compound noun, on the other hand, is applicable to several examples in the real world.
Rain forest - has two or more words and is a compound noun as each of the words add to its description. However there are several Rain Forests in the real world - another example is "Ice Cream".
The Empire State Building - has several words yet is a unique name for 1 building and so is a proper noun.