When switching from one to the other, we click on the cog wheel at the far right of the blue bar at the top of the Facebook page, right? And that's the correct way to move between your identities.
But when you switch identities and 'use Facebook as your Business Page', you no longer have access to all those friends associated with your Personal Profile - they're not attached to your business page.
Here's what you need to do.
Go ahead and share this with anyone you think is having trouble with 'Invite Friends' - when it happened to me it took a lot of hunting around, gnashing of teeth and Googling to find the answer!