It's curious that you don't want any error checking (in which case you could use Watch Long Processes Through CGI).

I can see two approaches.

• Spawn a task to do the work.

  use POSIX qw( setsid );
  
  print "Content-Type: text/plain\n";
  print "\n";
  print "1";
  
  my $pid = fork();
  exit(0) if !defined($pid) || $pid;
  
  open STDIN,  '</dev/null' or die;
  open STDOUT, '>/dev/null' or die;
  open STDERR, '>&STDOUT'   or die;
  setsid();
  
  ... handle request ...
  [download]

• Assign the work to an existing task or to a cron job.

  ... queue request in a file or database ...
  
  print "Content-Type: text/plain\n";
  print "\n";
  print "1";
  [download]