I hadn't realized until now that you can have F-Droid auto-update your apps(using the Privileged Extension) even on Android 5+. First you have to install this: https://f-droid.org/packages/org.fdroid.fdroid.privileged.ot...

It will download a zip file to /data/data/org.fdroid.fdroid/ota/ that you have to flash. After having flashed that file you can use the regular privileged extension to update the system app: https://f-droid.org/packages/org.fdroid.fdroid.privileged/