When you have a Hype document that scales proportionally in the height and width dimension, you need to set up a dynamically adjusting enclosing Div that corresponds to the ratio you desire. This excellent script by @h_classen is what you want to use. What that does is adjust the enclosing DIV which contains your responsive Hype document and keeps the ratios correct as your width is reduced. You'll still need the minimum height Class set, but this should help you avoid the bottom white space. The min-height will be overridden by the CSS set on the div since inline CSS overrides any class definitions you have referenced elsewhere.
You could also set the scene background to be the same background as your blog's background if this is not what you'd like to do.