i've got a HTC One X running XenonHD Rom (Android 5.1.1) and managed to download the HTC Car mode App from a third Party Website (apkmirror). The App has Bern installed without problem, also your xposed module decteted the app. But i'm not able to open the App.
Anyone a solution?
Anyway, what you're describing is an extreme case I'm not would work even without the mod.
The car app relies heavily on the HTC Sense framework, so installing it on a non-sense ROM would most likely not work.