I think this gets you a little closer:

use warnings;
use strict;
use XML::Twig;

my $twig = new XML::Twig(
    twig_handlers => { 'REF' => \&REF }
);
$twig->parsefile("AFP_ENG_20050316.0102.xml");    #Parse the file
$twig->print();

sub REF {
    my ( $twig, $field ) = @_;
    my $ref_text = $field->text();
    my $ref_ext  = $field->att('EXT');
    $field->set_text($ref_ext) if $ref_ext;
}
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You can try to use the cut method to remove elements.