You could definitely do this – instead of using the Hype Animations plugin, you would need to upload it as a regular Hype export and embed it within your theme. This is a good place to start: Video: Embedding in Wordpress (Posts, Pages & Themes)
The video on that post will show you how you can reference an uploaded Hype document, and was created prior to the Hype Animations plugin. While you can embed a Hype Animation shortcode within your theme using something like this:
<?php echo do_shortcode( '[hype_animations_anim="12"]' ); ?>
I think it is better to embed it outside so you have full control over the inline CSS styles for the menu and you can more easily position it.