In my opinion, your three examples don't read well because the beginning and the part after the dash don't connect together very well. The main clause of the first part is "We need laws", a positive statement, and the last part is "Not ... not ..." which gives negative examples. "that don't allow ..." in the sub-clause cannot have the same kind of strength as the main clause that can make the subsequent clause work in tandem with it. Your sentence can almost read like "We need laws - not political, not ...", which doesn't make much sense.
If you want a structure like your quoted sentence, you can probably say something like this (although it doesn't have exactly the same meaning).
- No violation of personal rights should be allowed/condoned by the law - no political, civil, or basic human rights violation.
I think your original sentence can best be said without the negatives ("that don't allow ..." seems to be a poor way of describing a clear and strong objective). I would state it more affirmatively.
- We need laws that prohibit/condemn/preclude any (kind of) violation of rights, be it [whether it is/be] political, civil, or basic human rights.