Assuming your first file is in a consistant format, you can trim off everything not in between quotes before saving to the hash. You could do your first while loop like this:

while (<F1>){
    s/\A
        [^"]*   # Read until we get to a quote char
        "       # The actual quote char
        ([^"]*) # Read (and save) until we hit another quote char
    /$1/x;
    $HASH{$_}++;
}
[download]

This will fail for embedded double-quotes, but most programs fail when that happens, anyway, so I wouldn't worry about it.

----
I wanted to explore how Perl's closures can be manipulated, and ended up creating an object system by accident.
-- Schemer

Note: All code is untested, unless otherwise stated