I feel how to handle this might be:
People are allowed to stop working and continue at same basic standard of living. They can try finding other people on the island who can help them, for example there will be atleast some psychologists who may want to help them.
The island is a project with multi-decadal time horizon for its goals, so ensuring everyone is doing okay is priority for everyone. If lots of people are unhappy it affects everyone else.
If someone is being highly disruptive to the atmosphere there could be a prison that provides same standard of living but places some restrictions on who they can meet and for what purpose.
Here too, the leaders don’t get to keep secret information. Everyone on the island is made aware of the condition of the “prisoners” and what help is being provided to them. And they can try to push for different conditions in a more democratic manner.
Are you worried more about a few people wanting to leave and being imprisoned instead? Or a large number of people wanting to leave and the whole project collapsing?