If you don't want your $container->{KEYS} entry to build up successively as you recurse through the structure, you'd have to remove the last entry every time you encounter the 'Leaf' recursion end-point.  The following would, for example, create the desired output:

...
sub walkHash
{
    my $h = shift;
    my $container = shift;

    foreach my $key (keys %$h)
    {
        if($key ne 'Leaf' && ref $h->{$key} )
        {
            $container->{KEYS} .= $container->{KEYS} ? "/$key" : $key;
            walkHash($h->{$key}, $container);
        }
        else
        {
            #Process hash under 'LEAF' key here.
            print Dumper($container);
            $container->{KEYS} =~ s#/[^/]+$##;   # remove last entry
        }
    }
}
...
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