A permission set is a collection of settings and permissions that give users access to various tools and functions in Salesforce. A permission set can be used to extend users' functional access without changing their profiles. You can assign permission sets to individual users or groups of users based on their business needs.
In this case, you can create a permission set with Read access to the Government ID field on the custom object and assign it to the Institutional Team users who need access to this field via reports. References:
https://help.salesforce.com/s/articleView?id=sf.perm_sets_overview.htm&type=5
https://help.salesforce.com/s/articleView?id=sf.perm_sets_create.htm&type=5