fileA.txt contains what you listed as File A in the question. I put the content of File B into __DATA__.

#!/usr/local/bin/perl -w
use strict;

my(%products) = ();

open(my $fh, '<', './fileA.txt') || die "Could not read fileA.txt: $!\
+n";
while(<$fh>){
    chomp;
    my($key, $value)=split(/\|/);
    $products{$key} = $value;
}
close($fh);

my(@groups)=();
{
    local $/;
    my $string = <DATA>;
    @groups = split(/(?=Group)/, $string);
}

foreach (@groups){
    my $name;
    if(m/Group Name: (.+)/){
        $name = $1;
    }else{
        print STDERR "Could not find group name\n";
        next;
    }
    my(@prods);
    if(m/Product Name:\n(.*)\Z/ms){
        my $prod = $1;
        @prods = split(/\n/, $prod);
    }else{
        print "$name\n";
        print STDERR "Group $name didn't have any products\n";
        next;
    }
    print "Group ".$name."\n";
    my $total = 0;
    foreach (@prods){
        print "$_ $products{$_}\n";
        $total += $products{$_};
    }
    print "Total ".join('+', @prods)." = $total\n";
}

__DATA__
Group Name: Small
Junk: lll
Product Name:
AAA
BBB
Group Name: Big
Junk: ggg
Product Name:
AAA
CCC
DDD


#output
Group Small
AAA 10
BBB 20
Total AAA+BBB = 30
Group Big
AAA 10
CCC 30
DDD 40
Total AAA+CCC+DDD = 80
[download]

If the File B like:
Group Name: Small
Junk: lll
Product Name:
AAA "This is test1 for me."
BBB "This may be I like it."
Group Name: Big
Junk: ggg
Product Name:
AAA "This is t2."
CCC "This should be ok"
DDD "This is the last one. we shoud have more."


How do I split the string of each product? The output is same as before.

If your goal is to discard everything following "AAA" or "BBB", and so on, you could modify the split statement to look like this:

@prods = split(/\s[^\n]+\n/, $prod);
[download]

This will remove everything following the first whitespace character in each of the lines following "Product Name:". Is that what you were trying to do?

May the Force be with you

my(%file_a);
open(FILEA,"path_to_file_a");
open(FILEB,"path_to_file_b");

while(my $line = <FILEA>){
   chomp($line);
   my @values = split /|/, $line;
   $file_a{$values[0]}=$values[1];
}
close(FILEA);
my ($calc_prod,$total);
while(my $line = <FILEB>) {
    chomp($line);
    if($line =~ m/^Group Name/) {
        print "Total: $total\n" if $total > 0;
        print "Group ", {$line =~ m/^Group Name: (.*)$/}, "\n";
        $calc_prod = 0;
        $total = 0;
    }
    next if $line =~ m/^Junk/;
    if($line =~ m/^Product Name/) {
        $calc_prod = 1; next;
    }
    if ($calc_prod) {
        print "$line ",$file_a{$line};
        $total += $file_a{$line};
    }
}
close(FILEB);
[download]