Kubo Formula for Quantum Hall - Derivation and Errors(?)

When one computes Hall conductivity $sigma_{xy}$, one can show that the zero temperature Kubo formula gives

begin{align}
sigma_{xy}(omega) = -frac{i}{omega} sum_{nneq 0} left[frac{langle 0|J_y |nrangle langle n | J_x|0 rangle}{hbar omega + E_n – E_0} – frac{langle 0|J_x |nrangle langle n | J_y|0 rangle}{hbar omega + E_0 – E_n}right]
end{align}

Now, one is in many instances interested in the $omegarightarrow 0$ limit, in which case one Taylor expands the denominators:

begin{align}
frac{1}{hbar omega + E_n – E_0} = frac{1}{E_n – E_0} – frac{hbaromega}{(E_n – E_0)^2} + mathcal{O}(omega^2)
end{align}

In particular, in many places (e.g. David Tong’s Quantum Hall notes) they argue away the $omega^0$ term so that there is no zero-frequency divergence. I want to know how this is explicitly done.

Another stackexchange post argued for the use of the trick $J_x sim v_x = frac{partial}{partial t} x = [H,x]$, which cancels the energy factors in the denominator and result in the desired vanishing of leading term because the commutator $[x,v_y]=0$. The problem, however, is that performing such a trick for the second term on both $J_x$ and $J_y$ gives a term proportional to the commutator $[x,y]=0$. Thus, such an argument also shows that the second order term also vanishes, which is patently false.

To argue that the leading term vanishes, Tong’s lecture notes invoke parity and gauge-invariance (without explicit argument). Parity makes sense, but the fact that $sigma_{xy} = -sigma_{yx}$ should be a result, not a constraint, so I feel there should be a more direct way of showing that it vanishes. The statement about gauge-invariance is mysterious to me.

Why is the Heisenberg equation of motion approach to this wrong, and what is the proper way to explicitly show that the leading term vanishes?