Not really that hard. Convert your from and to dates to something you can sort on (YYYY-MM-DD), then pattern match each line, convert the date there as well, and compare. When testing this, make sure to call the Perl script with the from and to dates as arguments and in the format specified by the OP.

use strict;
use warnings;

{
my %mon = (
    'JAN' => 1,  'FEB' => 2,  'MAR' => 3,  'APR' => 4,
    'MAY' => 5,  'JUN' => 6,  'JUL' => 7,  'AUG' => 8,
    'SEP' => 9,  'OCT' => 10, 'NOV' => 11, 'DEC' => 12
);
sub convDate {
    my $d = [split /-/, $_[0]];
    return sprintf('%04d-%02d-%02d',
        $d->[2], $mon{uc $d->[1]}, $d->[0]);
}}

my $from = convDate($ARGV[0]);
my $to = convDate($ARGV[1]);
my @d = <DATA>;

for (@d) {
    if (m/(\d+-\w+-\d+)_\d+\.\d+\.\d+/g) {
        print if convDate($1) ge $from &&
                 convDate($1) le $to;
    }
}

__DATA__
ABC01, 91XYZ889=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 02-Dec-2011_
+00.34.51, bigFatLog_02-Dec-2011_00.34.06.log
ABC03, 93XYZ272=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 07-Dec-2011_
+09.21.58, bigFatLog_07-Dec-2011_09.20.57.log
ABC02, 93XYZ807=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 08-Dec-2011_
+23.00.15, bigFatLog_08-Dec-2011_22.59.34.log
ABC05, 91XYZ525=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 10-Dec-2011_
+10.01.36, bigFatLog_10-Dec-2011_10.01.00.log
ABC01, 93XYZ252=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 12-Dec-2011_
+11.58.23, bigFatLog_12-Dec-2011_11.57.20.log
ABC03, 93XYZ543=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 12-Dec-2011_
+23.34.07, bigFatLog_12-Dec-2011_23.33.23.log
ABC04, 92XYZ066=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 13-Dec-2011_
+01.00.31, bigFatLog_13-Dec-2011_00.59.29.log
ABC05, 93XYZ184=_=SOMEBODY.NAME@DOMAIN.COM, HighPriority, 13-Dec-2011_
+01.54.41, bigFatLog_13-Dec-2011_01.54.04.log
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