This is actually two replies in one. First, thanks (and ++) to Corion for pointing out the mistake in my solution above.
Now Bindo, your question for a solution without CPAN... I think doing SFTP in pure Perl, while possible, would be difficult and error-prone. If you just don't want any additional CPAN modules, that's another question:
As far as I see, your $REM_FILE already is a file name without directory, so you don't need the basename function and module, and can simply
$sftp->get($REM_FILE, $REM_FILE, callback => \&show_progress)
Perhaps that already does what you want.
second parameter in get