I think I would use a lightweight 'reference' table with the unique index, put the actual data in the partitioned table, and use a foreign key to enforce a 1:1 relationship.

Something like this:

    create table orders(
             order_id int unique
           , year int
           , primary key (order_id, year)
    );

    create table order_data(
             order_id int
           , year int
           , LOADS_OF_COLUMNS_HERE jsonb

           , primary key (order_id, year)
           , foreign key (order_id, year) references orders(order_id, year)
    ) partition by list(year);

Unique constraints work as long as the partition key is included in the constraint (which makes sense, so a table knows it’s the only partition for a value and checks it’s unique index).

I wonder if this worth the price: an extra (thing_uniq) table and extra key column on partitioned tables. So pay an extra query to the thing_uniq table before insert to target table. thing_uniq has one indexed column (the key). But we lose upsert.