Just an off-the-wall, out-of-the-box idea. Both col1 and col2 must be null to qualify precludes using an index, perhaps building a psudo index might be an option. This index would of course be a regular table but would only exist for a short period. Additionally, this relieves the lock time worry.
create table indexer (mytable_id integer primary key);
insert into indexer(mytable_id)
select mytable_id
from mytable
where col1 is null
and col2 is null;
The above creates our 'index' that contains only the qualifying rows. Now wrap an update/delete statement into an SQL function. This function updates the main table and deleted the updated rows from the 'index' and returns the number of rows remaining.
create or replace function set_mytable_col2(rows_to_process_in integer)
returns bigint
language sql
as $$
with idx as
( update mytable
set col2 = 'ABCD'
where col2 is null
and mytable_id in (select mytable_if
from indexer
limit rows_to_process_in
)
returning mytable_id
)
delete from indexer
where mytable_id in (select mytable_id from idx);
select count(*) from indexer;
$$;
When the functions returns 0 all rows initially selected have been processed. At this point repeat the entire process to pickup any rows added or updated which the initial selection didn't identify. Should be small number, and process is still available needed later.
Like I said just an off-the-wall idea.
Edited
Must have read into it something that wasn't there concerning col1. However the idea remains the same, just change the INSERT statement for 'indexer' to meet your requirements. As far as setting it in the 'index' no the 'index' contains a single column - the primary key of the big table (and of itself).
Yes you would need to run multiple times unless you give it the total number rows to process as the parameter. The below is a DO block that would satisfy your condition. It processes 200,000 on each pass. Change that to fit your need.
Do $$
declare
rows_remaining bigint;
begin
loop
rows_remaining = set_mytable_col2(200000);
commit;
exit when rows_remaining = 0;
end loop;
end; $$;