For those looking for an in-transaction solution, the following seems to work.
Instead of an `ENUM`, a `DOMAIN` shall be used on type `TEXT` with a constraint checking that the value is within the specified list of allowed values (as suggested by some comments). The only problem is that no constraint can be added (and thus neither modified) to a domain if it is used by any composite type (the docs merely says this "should eventually be improved"). Such a restriction may be worked around, however, using a constraint calling a function, as follows.
START TRANSACTION;
CREATE FUNCTION test_is_allowed_label(lbl TEXT) RETURNS BOOL AS $function$
SELECT lbl IN ('one', 'two', 'three');
$function$ LANGUAGE SQL IMMUTABLE;
CREATE DOMAIN test_domain AS TEXT CONSTRAINT val_check CHECK (test_is_allowed_label(value));
CREATE TYPE test_composite AS (num INT, word test_domain);
CREATE TABLE test_table (val test_composite);
INSERT INTO test_table (val) VALUES ((1, 'one')::test_composite), ((3, 'three')::test_composite);
-- INSERT INTO test_table (val) VALUES ((4, 'four')::test_composite); -- restricted by the CHECK constraint
CREATE VIEW test_view AS SELECT * FROM test_table; -- just to show that the views using the type work as expected
CREATE OR REPLACE FUNCTION test_is_allowed_label(lbl TEXT) RETURNS BOOL AS $function$
SELECT lbl IN ('one', 'two', 'three', 'four');
$function$ LANGUAGE SQL IMMUTABLE;
INSERT INTO test_table (val) VALUES ((4, 'four')::test_composite); -- allowed by the new effective definition of the constraint
SELECT * FROM test_view;
CREATE OR REPLACE FUNCTION test_is_allowed_label(lbl TEXT) RETURNS BOOL AS $function$
SELECT lbl IN ('one', 'two', 'three');
$function$ LANGUAGE SQL IMMUTABLE;
-- INSERT INTO test_table (val) VALUES ((4, 'four')::test_composite); -- restricted by the CHECK constraint, again
SELECT * FROM test_view; -- note the view lists the restricted value 'four' as no checks are made on existing data
DROP VIEW test_view;
DROP TABLE test_table;
DROP TYPE test_composite;
DROP DOMAIN test_domain;
DROP FUNCTION test_is_allowed_label(TEXT);
COMMIT;
Previously, I used a solution similar to the accepted answer, but it is far from being good once views or functions or composite types (and especially views using other views using the modified ENUMs...) are considered. The solution proposed in this answer seems to work under any conditions.
The only disadvantage is that no checks are performed on existing data when some allowed values are removed (which might be acceptable, especially for this question). (A call to `ALTER DOMAIN test_domain VALIDATE CONSTRAINT val_check` ends up with the same error as adding a new constraint to the domain used by a composite type, unfortunately.)
<strike>Note that a slight modification such as `CHECK (value = ANY(get_allowed_values()))`, where `get_allowed_values()` function returned the list of allowed values, would not work - which is quite strange, so I hope the solution proposed above works reliably (it does for me, so far...).</strike> (it works, actually - it was my error)