I had the same problem and was finally able to solve it!
You have to change the two apps “microg Services Core” and “microg Services Framework” to become System Apps.
In order to do so you need to move their app folders from /data/app to /system/app
You can do this with ES Explorer for example. The folders of the app are called “com.google.android.gms-2” and “com.google.android.gsf-1”.
Then you have to change their permissions as its written here:
After a restart of the phone it worked again for me.
Note: I originally installed the microg system via FDroid. I dont know if there are differences if you did install it manually.
I had to reinstall microG from F-Droid after updating to Fairphone Open 19.02.1. After that I again had to make them system apps. This time I used Amaze from F-Droid to move them and change permissions which also worked just fine.