That is, are the only field configurations which give a non-zero winding number ones in which the Fourier transform includes a factor like $\theta(k^0)\hat{D}\delta(k^2)$, where $\hat{D}$ is some differential operator acting on the delta function?
Below I will explain why I think this may be the case, and hopefully we can have a discussion about it:
Say we consider the theory of a massless fermion $\psi$ coupled to some gauge field $A^a_{\mu}$ in $4$ spacetime dimensions. The Lagrangian I would like to consider is $$L=i\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-\frac{1}{4g^2}f^a_{\mu\nu}f^{a\mu\nu}$$We know that this theory has a chiral anomaly of the form $$\partial_{\mu}J^{\mu}_5=-\frac{C}{32\pi^2}\varepsilon^{\mu\nu\rho\sigma}f^a_{\mu\nu}f^a_{\rho\sigma}$$ Where $\text{tr}(t_at_b)=C\delta_{ab}$ and $t_a$ are the generators of the gauged Lie group, and the representation $\psi$ transforms under. Note first of all that if we integrate both sides we get
$$Q_5(t=\infty)-Q_5(t=-\infty)=-2C\nu$$ Where $\nu$ is the instanton winding number.
At this point, I would now like to consider the Wilsonian Effective Lagrangian $L_{\Lambda}$, for which all of the modes of mass $m>\Lambda$ are integrated out. Under a chiral transformation $\psi\to e^{i\alpha\gamma_5}\psi$, we know that $L_{\Lambda}\to L_{\Lambda}-\alpha\frac{C}{32\pi^2}\varepsilon^{\mu\nu\rho\sigma}f^a_{\mu\nu}f^a_{\rho\sigma}$.
For this to be the case, it would be essential that none of the massive modes of mass $m>\Lambda$ have non-zero winding number. This must be case, or else instantons would affect the transformation properties of $S_{\Lambda}=\int d^4xL_{\Lambda}$ under a chiral transformation.
This must also be the case for all values of $\Lambda$, since this chiral transformation does not depend on $\Lambda$ either. Therefore the only room we have for instantons is them to be massless, and have the factor mentioned above in their Fourier transform.
The statement I left in bold is what this argument hinges on, and is also a statement that I am not certain of it's validity. I know for sure that if we only integrated out the high mass modes of $\psi$, that would be the way $L_{\Lambda}$ transforms. However, the fact that we also integrate out the high mass modes of $A^a_{\mu}$ potentially screws this nice transformation up.
I would really appreciate getting both feedback on the argument, and whether or not the statement in bold is true!