My question is this: Is there a way that an anonymous sub stored as a hash value can have access to its associated key?

Unlike what several others have said, you can. In the code below, the anonymous sub gets the key belonging to the value passed in as its first argument.

#!/usr/bin/perl

use strict;
use warnings;

package Cute;

sub TIEHASH {bless {}, shift}
sub STORE   {my ($self, $key, $sub) = @_; 
             $$self{$key} = sub {$sub->($key)}}
sub FETCH   {${$_[0]}{$_[1]}}

package main;

tie my %hash, 'Cute';

my $greeting = sub {print "Hello, $_[0]\n"};

$hash{world} = $greeting;
$hash{earth} = $greeting;

foreach my $key (qw /world earth/) {
    &{$hash{$key}};
}
 
__END__
Hello, world
Hello, earth
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