Break it down to just the lines you want and fix those in isolation. SSCCE:

use strict;
use warnings;

use Test::More tests => 2;

my $input    = '</div></div><div><div class="blue"></div></div>';
my $expected = '</div></div>';
my $remove   = '</div><div><div class="blue"></div>';

like ($input, qr/$remove/, 'Match found');    # Just for illustration
$input =~ s/$remove//;
is ($input, $expected, 'Strip successful');
[download]

Searching has suggested this is silly to try without a cpan solution

Quite right. If you start to attempt to parse HTML in this fashion you are starting down a long and slippery slope which ends in the belly of the grue. Recant now while there is still time.