On the causal structure of spacetimes: piecewise $C^1$, $C^k$ or $C^\infty$?

The answer is that it doesn't matter, as long as the metric itself is sufficiently regular. In the following notes by Chrusciel, the basic assumption is that the metric is $C^2$:

[C] Chruściel, Piotr T., Elements of causality theory, arXiv:1110.6706 (2011).

I'm not up on the state of the art of how low you can push the regularity of the metric and keep all the standard results of causality theory, but in the following paper it is shown that it all works even for $C^0$ metrics (those that avoid what the authors call causal bubbles):

[CG] Chruściel, Piotr T.; Grant, James D.E., On Lorentzian causality with continuous metrics, arXiv:1111.0400 Classical Quantum Gravity 29, No. 14, Article ID 145001, 32 p. (2012). ZBL1246.83025.

The paths actually considered in these references are locally Lipschitz, which covers all the regularity classes that you've included in the question. Corollary 2.4.11 [C] shows that both the timelike and causal futures/pasts coincide, whether you use locally Lipschitz paths or piece-wise broken geodesics, respectively timelike and causal. If you deal with $C^\infty$ metrics, then your geodesics will also be $C^\infty$, so this result shows that it is indeed safe to stick with the piece-wise $C^\infty$ class. Then the result that you refer to, about deforming a non-null geodesic into a timelike curve with the same end points, is known as a push-up lemma in the literature. You can find this result for locally Lipschitz curves in Proposition 2.4.18 [C], which relies on the technical intermediate results Corollary 2.4.16 [C] and Lemma 2.4.14 [C].

Let me expand on Igor Khavkine's answer, especially:

"The answer is that it doesn't matter, as long as the metric itself is sufficiently regular."

In our recent paper: The future is not always open we clarified all these issues with the regularity of the curves vs. the regularity of the metric (and showed that there are some pathologies in low regularity). Maybe this is also of interest to you.