It worked for me; however, it wouldn't open LOGFILE on the first try, so I made a few minor adjustments.

#!/usr/bin/perl -l

BEGIN { $| = 1; }

use strict;
use warnings;
use Term::ReadLine;
use POSIX qw(strftime);

my $username = getlogin;
my $logpath  = '/tmp/sessionlogs';

my(%cfg) = (
    shell       => '/bin/bash',
    dateformat  => '[%Y%m%d-%H%M%S] ',
    logpath     => $logpath,
    logname     => "$username.session.log",
);

my(@prohibited_cmds) = (
    'mc',
    '/usr/bin/mc',
);

open LOGFILE, ">>$cfg{'logpath'}$cfg{'logname'}" 
  or die "Cannot open log file: $!";

print "enter 'exit' to quit.";
print LOGFILE "-----------------------------------\n";
print LOGFILE strftime( $cfg{'dateformat'}, localtime );
print LOGFILE "Session started for $username.";

my $term = Term::ReadLine->new('Shellwrapper');
$SIG{'INT'} = 'IGNORE';

while (defined ($_ = $term->readline('SH+log> '))) {
    if (/^\s*quit\s*$/ or /^\s*exit\s*$/) {
        print LOGFILE strftime( $cfg{'dateformat'}, localtime );
        print LOGFILE "$_\n";
        last;
    }
    print LOGFILE strftime( $cfg{'dateformat'}, localtime );
    print LOGFILE "$_\n";
    system "$cfg{'shell'} -c '$_'";
    print "\n";
}

print LOGFILE strftime( $cfg{'dateformat'}, localtime );
print LOGFILE "Session terminated for $username.";

close LOGFILE;
[download]