Important notice: When downloading the apk, watch out for which Android version it is intended... the one for Pie is from August 12th. I got the one from today, which is for Android 5.0. :doh:
EDIT once again: Now that I got the correct version, the error messages have stopped popping up. I didn't even have to delete the old folder this time. Just installed the apk, and fine I was.
As the battery was down to 7% in the meantime I am now recharging, and let's see how it goes from there. If the latest versin of Google Play services behaves and stops draining my battery. I'll report back on Friday or so.