WWW::Mechanize->new(
   autocheck  => 1,
   cookie_jar => $cookie_jar,
   default_headers => HTTP::Headers->new(
      User_Agent      => 'Mozilla/5.0 <snip>',
      Referer         => 'https://www.<snip>',
      Accept          => 'application/<snip>',
      Accept_Language => 'en-US,en;q=0.5',
      Accept_Encoding => 'gzip, deflate',
      Connection      => 'keep-alive',
   ),
   show_progress => 1,
);
[download]

Both the original version and this version look fine to me.

(User_Agent => ... and 'User-Agent' => ... can be used interchangeably in both snippets.)