As to *any way of proving [the derivative of the delta function] is not the difference of two positive distributions*: indeed,
$\delta’$ is not even majorized by a positive distribution. Here’s a direct argument.

If we had $\delta’\le u$, for a positive distribution $u\in\mathcal D’(\mathbb R)$, then, for every couple of test functions, $f,g$ in $\mathcal D(\mathbb R)$ with $f\ge g\ge0 $ we would have
$$u(f)\ge u(g)\ge g’(0),$$
which is absurd because, for a fixed $f$, $g’(0)$ is certainly not bounded among all non-negative test functions  below $f$.