Eventually, I discovered the FlashWidget class, which allows you to show Flash content within your PopCap based application.
I looked over the class, it seemed relatively easy to throw an flash game into the app, so I tried it.
Performance left quite a bit to be desired, and switching back and forth with other applications doesn't work, so either I'm using the class incorrectly (likely, as I spent a total of 5 minutes getting the thing working), or the class is borked.
However, with the performance degradation, I'm still thinking that a rewrite/port will do better than just throwing the flash game into the framework.
Plus, it doesn't really solve anything, as it still requires the flash runtime to work.
So, I'll do more experiments later, combining flash and popcap's framework. Stay tuned.