You have a lot of options. The simplest is probably to use Sqlite's ".dump" shell command to generate the SQL. There are a number of options or ways you can do this. You could use ".dump" which dumps all the schema and data, and then make any modifications to the dump-file and then load it into postgres.
To just dump row data for table "my_table" as CSV you can use:
sqlite> .headers on
sqlite> .mode csv
sqlite> .output model_data.csv
sqlite> SELECT * FROM my_table;
Then you can bulk load the CSVs using Postgres:
COPY table_name(col1, ...)
FROM 'model_data.csv'
DELIMITER ','
CSV HEADER;
Peewee can dump data in a variety of formats, so that's one option:
models = [User, Tweet]
for model in models:
outfile = '%s.csv' % model._meta.table_name
with open(outfile, 'w', newline='') as f:
writer = csv.writer(f)
row_iter = model.select().tuples().iterator()
writer.writerows(row_iter)
Another option would be to use a Postgres foreign-data wrapper. Then connect to psql db, attach sqlite database using fdw, and do a straight copy from one to the other using psql shell.
https://github.com/pgspider/sqlite_fdw