If this information is stored in custom fields you can make it local to the project.
Meaning when creating/editing the custom field if you do not tick the box to have the custom field accessible org wide then others should not be able to see it, more info here.
Then since you are on business plan try to work with custom fields or rules.
In your case actually you can set up rules to create subtasks and in there you can use variables to be added in the name. At the moment the only variables available are these:
You can achieve further customizations with Flowsana though which might be an option in your case.