I created a new Lakebase project to retrace all my steps. 

0- I reused my service principal on the workspace

1- installed databricks authentication extension: 

CREATE EXTENSION IF NOT EXISTS databricks_auth;

2-Added the lakehouse service principal to the lakebase project

SELECT databricks_create_role('{UUID}', 'SERVICE_PRINCIPAL');

3- Enabled Data API to get authenticator user created

4- Finally granted authenticator role to the service principal

GRANT "{UUID}" TO authenticator;

 And this time it worked. I do not understand what the difference is to last time, maybe my authenticator user was somehow corrupted.

Thank you @balajij8 & @szymon_dybczak for your answers and suggestions