You don't really need a trusted host relationship, you just need passwordless keys. If passwordless keys are impossible, then you will have to use the interactive method. Maybe you can go forward by hacking Net::Ssh::Perl to redirect the STDOUT part of the connection. Looking into the source of Net::SSH::Perl::SSH1, there seem to be handlers like SSH_SMSG_STDOUT_DATA and replacing the default handler with something that doesn't accumulate the string might help:

# Original code
sub cmd {
    ...
    unless ($ssh->handler_for(SSH_SMSG_STDOUT_DATA)) {
        $ssh->register_handler(SSH_SMSG_STDOUT_DATA,
            sub { $ssh->{_cmd_stdout} .= $_[1]->get_str });
    }
    ...
}
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I would try to supply my own callback like this:

my $ssh = Net::SSH::Perl->new(...);
open my $outfh, ">", $filename
  or die "Couldn't create '$filename' : $!";
binmode $outfh;
$ssh->register_handler('stdout', sub { print $outfh $_[1]; });
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If passwordless keys are at all possible, you could also consider File::Remote.

This is from the example:

use File::Remote qw(:replace);
   # read from a remote file
   open(REMOTE, "host:/remote/file") or die $!;
   print while (<REMOTE>);
   close(REMOTE);

   # writing a local file still works!
   open(LOCAL, ">>/local/file");
   print LOCAL "This is a new line.\n";
   close(LOCAL);

   # Read and write whole files
   my @file = $remote->readfile("host:/remote/file");
   $remote->writefile("/local/file", @file);
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0xbeef