and an attempt to get the data is made so: ...

Why do you need to get at the data in $message by reading from the associated filehandle? (seems like a rather cumbersome way to copy a string from one variable to another... :)

But if you really insist on doing that way, I suppose you have to reset the filepointer to the beginning before reading, e.g.

use strict;
use warnings;

open my $msg, "+>", \my $message or die "no message $!";

print $msg " foo\n";
print $msg " bar\n";

print "\$message:\n$message";

seek $msg, 0, 0;  # reset filepointer
my $text;
while (my $line = <$msg>) {
    $text .= $line;
}

print "\$text:\n$text";
[download]

Output:

$message:
 foo
 bar
$text:
 foo
 bar
[download]

In other words, once you have the mail in $message, why not pass it directly to Mail::Sendmail...