I've heard back from Ellen Hutson (their customer service is amazing!) and they said that because of the details of that die, additional pressure may be needed, and could I please send them a picture so they can try to figure out a solution. It sounds like I just need to find the right type of shim. The truth is that it is a cute die and it looks great even simply embossed. I'm satisfied.
I am, however, thinking it might be time to update my die cutter. Maybe.