At this current point in time it is not possible to set a Next-Gen access property (the one you can set in Project Settings > Details) via REST API, neither when creating the project nor afterwards.
There is an internal endpoint that you can currently use to change the project access setting but this is not documented, not supported and it can change without prior notice: /rest/internal/simplified/1.0/projects/PROJECT-ID/summary
In order to change the access settings you have to sent a PUT request to the /rest/internal/simplified/1.0/projects/PROJECT-ID/summary specifying:
accessLevel: "OPEN", "PRIVATE" or "LIMITED"
avatarId: avatar id of the project
key: project key
name: project name
- Call https://XXXXX.atlassian.net/rest/api/3/project endpoint to get project info:
- change the access settings to the above project by sending a PUT request to https://xxxxxxx.atlassian.net/rest/internal/simplified/1.0/projects/10303/summary: