Do you know if API service principal / user has USAGE on the database itself? This seems like the most likely issue based on information on the question. 

Quick Fix Checklist:

  Run these commands in order (replace api_user with the actual user from step 1):

  -- 1. Find the API user

  SELECT current_user();  -- Run this through your API first!

  -- 2. Grant database usage

  GRANT USAGE ON DATABASE your_schema TO `api_user`;

  -- 3. Grant function select (you already did this)

  GRANT SELECT ON FUNCTION your_schema.function_name TO `api_user`;

  -- 4. Find and grant access to underlying tables

  SHOW CREATE FUNCTION your_schema.function_name;  -- Check what tables it uses

  GRANT SELECT ON TABLE your_schema.table1 TO `api_user`;