muhomorr does a lot of the GmsCompat work for GrapheneOS. They aren't active on the discussion forum often, so hopefully I can help.
It seems the issue is with Google Play updating Google Play Services. What happens is Play installs a slightly modified Play Services but they don't change the version number. Then, when you try to install Play Services via the Apps app, it conflicts with the modified version installed in another profile.
Hopefully that makes sense.
I don't know if the issue is fixed yet, but here's a related commit on Github.
If it is fixed, though, we have to wait for an update to Google Play Services in the Apps app before we can start installing GMS in other profiles via Apps.