You can't install just "Google Play services" or just "Google Play Store", you have to install both at the same time (at least the Graphene App Store started to install both no matter which one I manually installed).
So no matter which one you actually need for the "Play Integrity API" feature, you need to install both.
The settings for "Play Integrity API" are in the app that requires it. You won't find it the general settings.
So what I have done is to install "Google Play services" and "Google Play Store" to the Private Space, as well as any apps that require "Play Integrity API".