Have you tried the "OsmAnd~" version on fdroid? I think that one is basically the full version without a paywall. You could try yourself if you find it worth it, and buy the official app if you want to support the project.
Yeah that's what im using from fdroid but Android auto is not available unless i subscribed or pay for the app. Wondering hows the app for people who already used it.
Basically you construct an Intent ACTION_INSTALL_PACKAGE with data pointing to the APK file and the extras EXTRA_NOT_UNKNOWN_SOURCE=true and EXTRA_INSTALLER_PACKAGE_NAME="com.android.vending" which tells the installer that this APK is not sideloaded and it's the play store asking to install it.
You might still need to enable unknown sources in Android Auto developer settings (separate from phone developer settings).
If I remember, I'll try to pull the code for my app from my PC and post it.