Unfortunately, there is a misunderstanding on how this feature works. It does not log tasks for organizations. It is meant for internal reference for people working on the same tasks.
Limiting the dropdown will not do what you want as it will not limit access to tasks or make them available to different people.
The REI3 Tasks app was built for open collaboration between team members. It could be extended to address your use case, but it would take a lot of work as it would require a new permission system.
In this case it would be much less work to create a new app that was designed for this use case. A simple task managing app that would work as you describe would take maybe 6-8 hours. Redesigning the REI3 Tasks app to also address your use case would take a couple of days.