Force some composite type fields to be unique?
First of all, you can't add constraints to types therefore the type definition you've provided will error out.
Domains as you've already observed won't work since they can only accomodate a single type at a go rather than a composition of types.
You also don't have to add a not null constraint to a primary key - it's redundant, Postgres by default disallows null values for primary key columns.
The best approach will be to normalize the table asset so that you can maintain the restrictions you require since both domain definitions and type defintions can't accomodate these restrictions.
For the sake of completion, here's a suggestion for the table defintions.
Given that an asset has multiple images but an image can only belong to one asset, a one-to-many relationship captures this perfectly as shown below. Furthermore, given how your code is currently structured, different assets can have the same (lang, size) combo. Therefore, the unique constraint is added across both the asset_id, lang and size
create table asset (
  id bigserial primary key,
  /* other asset columns ... */
);

create table img (
    asset_id bigint references asset(id) not null,
    lang varchar(2) not null,
    size varchar(32) not null,
    img_path varchar(255) not null,
    unique(asset_id, lang, size)
);

Note, the unique constraint creates an index hence the checks will be fast. Therefore, as expected, the following insert fails:
insert into img
values 
    (1, 'en', '32 kb', 'bar path'),
    (1, 'en', '32 kb', 'foo path');