BUG #18249: pg_dump/pg_restore single schema with function1 calling function2

The following bug has been logged on the website:

Bug reference: 18249
Logged by: Wilma Wantren
Email address: wilma(dot)wantren(at)web(dot)de
PostgreSQL version: 13.4
Operating system: All
Description:

The company I work for used to only use Oracle databases for their program
and now also supports PostgreSQL.
With Postgres, we create a database, a user and a schema, which all have the
same name. The name is chosen by the customers.
Nevertheless, the customers would like to export the data and definitions
and import them under a new name.
This works well by using the -O (no owner) option for pg_dump/pg_restore and
only exporting and importing the schema. After the import the schema is
renamed. So far so simple and easy!
But there is one problem: we have a database function function1, which calls
another function function2. The call is made without a schema prefix because
these are just two functions that we make available to the customers and we
don't think an extra schema for these two functions is worthwhile. However,
function1 is used in an index and the restore of the index fails if there is
data in the corresponding table column. This is because pg_dump/pg_restore
set the search_path to '' when restoring, which means that function1 cannot
find the function2.
Is there any way to solve this? It is not a big problem and we can tell the
customer to ignore the error message and recreate the indices after the
import, but it would be nicer without an error message.

Here are the steps to reproduce:
Contents of file "create_dumper.sql":
create user dumper;
create database dumper owner dumper;
alter user dumper set search_path to dumper;
\c dumper dumper;
create schema dumper;
create or replace function function2(str varchar) returns varchar as $$
begin
return str || str;
end;
$$ language plpgsql strict immutable;

create or replace function function1(str varchar) returns varchar as $$
begin
return str || 'hallo' || function2(str);
end;
$$ language plpgsql strict immutable;

create table table1(str varchar not null);
insert into table1 values('test');
create index index1 on table1(function1(str));

Call with
psql -f create_dumper.sql

Export:
pg_dump -h localhost -p 5432 -U dumper --schema dumper -O -f dumper.sql

Contents of file "create_restorer.sql":
create user restorer;
create database restorer owner restorer;

Call with:
psql -f create_restorer.sql

Import:
psql restorer restorer -f dumper.sql