Rather general versions of these theorems have been established for "sets of finite perimeter." You can find a recent paper on this subject here:

<cite authors="Chen, Gui-Qiang; Ziemer, William P.; Torres, Monica">_Chen, Gui-Qiang; Ziemer, William P.; Torres, Monica_, [**Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws**](https://doi.org/10.1002/cpa.20262), Commun. Pure Appl. Math. 62, No. 2, 242-304 (2009). Available at [arXiv:0709.3673](https://arxiv.org/abs/0709.3673) [math.AP]. [Zbl&nbsp;1158.35062](https://zbmath.org/?q=an:1158.35062).</cite>

On the other hand, you say you don't want "heavy machinery," so it is hard to guess what you are looking for.