Copying a PostgreSQL schema and its data from one database to another

I am trying to copy a PostgreSQL schema and it's data from one database to another without impacting the availability of the current schema (old_schema). I would also like to do this for a specific subset of tables within the schema, and want the new schema to have a different name in the other database.

I am using Python for steps 1. and 2. in the following procedure,

1. Get a list of table names from old_schema that I want to copy.

    select
        distinct
        information_schema.columns.table_name as table_name
    from
        information_schema.columns
    where
        information_schema.columns.table_schema = 'public'
        and
        information_schema.columns.table_name ~ 'lime_.*'
    ;

2. Loop through the table names, creating the same tables in the new_schema

    create table if not exists {new_schema}.{lime_table} (like {old_schema}.{lime_table} including all);

and also copy the data over from the old_schema to the new_schema for each table

    insert into {new_schema}.{lime_table} (select * from {old_schema}.{lime_table});

3. Now we have a copy of the tables we want in new_schema.

Here comes the part with unexpected behavior from PostgreSQL. For migrating the new_schema to the other database, we first dump it to a file using

    pg_dump.exe
        --host="<HOST>"
        --port=<PORT>
        --username=<USERNAME>
        --table="lime*"  // redundant because of step 1.
        --format=c
        --schema=new_schema // our `new_schema`
        --exclude-schema="public" // doesn't work, public still being written in dump file
      "<DB_NAME>" > C:\Users\<PATH>\backup.sql

However, even after copying tables from public to new_schema in step 2., specifying to pg_dump to only dump new_schema, and also specifying to exclude the public schema (the schema where the data originates from), we still get public.<table> in the dump file! It's just like what's outlined in this question - pg_dump does not honor -n.

4. If the dump were to work, the plan is to use the following to copy new_schema to a different database.

    pg_restore.exe
        --host="<HOST>"
        --port=<PORT>
        --username=<USERNAME>
        --verbose -1
        --dbname="<DB_NAME>"  
      C:\Users\<PATH>\backup.sql

The versions of PostgreSQL I am using are outlined in the dump files.

-- Dumped from database version 10.9.17

-- Dumped by pg_dump version 14.1