You can fetch category-related data via REST API by rest/V1/categories/:categoryId in Magento 2.
The Response of the API will be all the necessary data related to the Category with custom attributes also.
Category API will respond only if you have resource permission to access the given role.
Magento_Catalog::categories
How to get category data by REST API in Magento?
Action: GET
Request URL: <WEBSITE_URL>/rest/V1/categories/:categoryId
Where replace categoryId with your original category Id.
Headers: Authorization:Bearer <ADMIN_TOKEN>
In the Headers, you need to replace <ADMIN_TOKEN> with your original token value for the permitted store owner.
Payload: None
Response: Category Data array.