You can definitely have PG ignore rows that would violate constraints by specifying "ON CONFLICT DO NOTHING"[1] as part of your INSERT statement.
If you wanted to capture those rows on the fly, one way would be to perform the load via a PL/PgSQL program and include an exception handler specific to the particular constraint violation(s) you expect to hit. But I guess then you'd have to load them one row at a time...and that's not too different from just doing that in your high-level language of choice. Also the PL/PgSQL docs specifically recommend against this idea.[2]
[1]: https://www.postgresql.org/docs/11/sql-insert.html#SQL-ON-CO...
[2]: https://www.postgresql.org/docs/11/plpgsql-control-structure...
EDIT: formatting and clarity