Probably your best bet is to use MIME::Lite to construct the message, then express the entire message as a string. Once it is in string form, you can have Mail::DKIM create a signature for it. Concatenate the signature and the original message, then send it using something other than MIME::Lite.
If you really want to use MIME::Lite to send the message, perhaps something like this will work...
# $msg is the MIME::Lite object
### Prepare the Mail::DKIM object
use Mail::DKIM::Signer;
my $dkim = Mail::DKIM::Signer->new(
Algorithm => "rsa-sha1",
Method => "relaxed",
Domain => "myhost.com",
Selector => "mx1",
KeyFile => "./private.key",
);
# Mail::DKIM wants the message in the form it is transmitted, i.e. CR-LF line endings
my $raw_data = $msg->as_string;
$raw_data =~ s/\n/\015\012/gs;
$dkim->PRINT($raw_data);
$dkim->CLOSE;
# Now create the signature
my $sig = $dkim->signature;
my ($header_name, $header_content) = split /:\s*/, $sig->as_string, 2;
# Insert the signature at the beginning of the message (uses an undocumented API of MIME::Lite)
unshift @{$msg->{Header}}, [ $header_name, $header_content ];
# Now send the message
$msg->send();
