I figured out a solution to get it working. I should have thought of it earlier.
I uninstalled the amazonmp3-unsigned.apk system app using ES File Explorer. I then used Google Play to install the new com.amazon.mp3 APK as a user app. Somehow I had gotten it stuck in my head that it would need to be a system app to work with the rest of the Kindle software. Duh.
It seems to work perfectly. I can access Prime music just fine, and it seamlessly connects to my Amazon account. The only downside is that the app from the Play store is designed for phones and locks the app into portrait mode, whereas the native Kindle version supports landscape. But, hey, that's a small price to pay.