Beyond using the Schwartzian transform, you can use the magic of pack to create your sort keys! Keys N, n and C can all work, if your IDs are nonnegative and small enough (0xFFFFFFFF for the first [which will anyway put you in trouble when you try to manipulate them, or 0xFFFF for the second, or a measly 0xFF for the third).

sub make_key {
  my $x = shift;
  pack 'N*', @{$x->{Path}};
}
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is enough to create keys; you compare these keys with cmp, so you don't need to pass a coderef to sort (yay!).