If anyone is searching for a solution, here is the one that worked for me. You need to use Postgres endpoint to load data into Babelfish.

Just connect to Postgres enpoint of Babelfish and look at the schema of babelfish_db database. You need to create AWS DMS tasks 2 rules that will add:

• convert destination schema to lowercase
• add prefix to schema: your_db_name_ -> "_" at the end is important.\

Hope this will work for you.