What you want to set are the headers of the request you send, as that is what the server then sees. The following untested code should set the headers of the request:

$request = HTTP::Request->new('GET', $inURL);
$request->header( 'Accept-Content' => '*/*' );
$request->header( 'Accept-Language' => 'en' );
$request->header( 'Connection' => 'keep-alive' );
$request->header( 'Extension' => 'Security/Remote-Passphrase' );
$request->header( 'UserAgent-CPU' => 'PPC' );
$request->header( 'UserAgent-OS' => 'MacOS' );
$request->header( 'Transfer-Encoding' => 'deflate,gzip;q=0.3' );
[download]

This should be the correct headers, but I guessed on many of them, so some more experimentation is in order.

Also, you're specifying a Connection: Keep-alive header, which won't work properly with the standard uses of LWP, as LWP dosen't support keep-alive connections unless you also make use of LWP::ConnCache. So after some initial experimentation, you might want to change that header to Connection: close, which shouldn't affect the rest of the operation.

While browsing the documentation of LWP and LWP::ConnCache, it seems that LWP tries to Do The Right Thing if you simply specify a Connection: keep-alive header, so it might Just Work™