Whilst reading about differential characters, I found the following statement in the original paper by Cheeger-Simons "Differential Characters and Geometric Invariants" and later papers (the quote is taken from "Cheeger-Simons differential characters with compact support and Pontryagin duality" by Becker et al.):
Differential forms must vanish if they take values in a proper subring of the reals upon integration over any chain. In particular, differential forms having integral periods must be closed.
Becker et al. state that this can be proven by a "rescaling argument". However, I don't really know what is meant by this.
UPDATE: The argument that I cooked up was, as correctly pointed out by @user954180, faulty from the start. For this reason it was removed.
I don't follow your argument: $d\omega$ is well-defined, not well-defined up to a scalar. It seems to me the argument uses two points.
(1) Suppose a differential $(k+1)$-form $\eta$ is nonzero at $p$. Let $c = c(\epsilon): \Delta^{k+1} \to M$ exponentiate a simplex of diameter $\epsilon$ in $T_p M$, so that $c^* \eta_p$ is a nonvanishing top form on $T_p \Delta^{k+1} = \Bbb R^{k+1}$. Then $c^* d\omega = c^* (d\omega)_p + O(\epsilon)$. It follows that $\int_{c(\epsilon)} d\omega \approx c\epsilon^{k+1} + O(\epsilon^{k+2})$ for some $c \neq 0$. It follows that if $\eta$ is nonvanishing, there is some chain on which $|\int_c \eta| \in (0,1)$.
(2) However, because $\int_c d\omega = \int_{\partial c} \omega \in \Bbb Z$, it follows that $d\omega$ is vanishing.
I think (1) is the "rescaling argument".