Unfortunately, DMS is quite limited for Postgres. We trialled it but found these two limitations deal breakers:

* "AWS DMS doesn't support change processing on Amazon RDS for PostgreSQL. You can only do a full load."

* "AWS DMS doesn't map some PostgreSQL data types, including the JSON data type. The JSON is converted to CLOB."

http://docs.aws.amazon.com/dms/latest/userguide/CHAP_Source....

Hopefully it improves soon, since it's an otherwise awesome tool.