Scott
Unlike with user profiles, apps available in owner profile can't be installed in Private Space via Settings> System> Users> Profile Name> Install available apps
No option to move app into private space
On stock OS this only seems to open the Private Space's Play Store:
All this does on stock is launch the play store in the private space. You still have to install it separately. We removed that button because it was confusing people.
Simply install the app stores you want in the private space and install the apps as you would normally.
source: https://discuss.grapheneos.org/d/16670-private-space-on-android-15-grapheneos/23