Postgres dumping specific non-table objects
I could run a whole-database dump with -T options set to exclude all the tables I don't want. 


So what you want is to automatically exclude all tables but the ones you specify? You could always do this with dynamic sql..

SELECT format(
  'pg_dump %s',
  string_agg(  format('-T ''%I.%I''', schemaname, tablename), ' '  )
)
FROM pg_tables
WHERE schemaname = CURRENT_SCHEMA -- whatever schema.
  AND tablename NOT IN ('myDesiredTableName');


Run that with psql -c and then run the output from it to dump the database.