Instead of passing a direct reference to the handler sub, you can wrap it in a closure:

    twig_handlers => {
        mybutton => sub { $self->_twig_handle_button(@_) },
    },
[download]

Then your handler sub becomes a full-fledged object method:

sub _twig_handle_button {
    my ( $self, $t, $section ) = @_;

    my $parsed_href = {};

    $self->_handle_parsed_href( $parsed_href );
}
[download]