Wait, if there was no header, it should die (die "Bad response status line: '$status'" unless $laxed;)

The status line is considered part of the header. A minimal header would be:

HTTP/1.0 200 OK<CR><LF>
<CR><LF>
body
[download]

Is the status line missing? Is the blank link missing?

Anyway, you'll probably have to drop LWP and create your own socket.