Maybe this was included in the getting started guide, but I completely missed it.
My CODAWheel was configured in "selfie-stick" mode out of the box, so pressing the button just adjusted the volume on my phone, even from within the Bridget app.
In order to get it to work properly with the Bridget demo app, I had to first download the "CODAWheel" app from the iOS App Store, then manually select one of the alternative actions.
It seems like once I made a manual selection one time, the next time I tried the Bridget app, it successfully put the wheel in "Compatible App" mode.
Not sure if this is a bug with the Bridget app or a hardware/firmware issue with the CODA wheel, but that initial manual configuration step made it work for me!
Hope this helps someone out there.