How do I derive this Ward-type identity?

I am trying to derive a Ward-type identity between amplitudes involving $barpsi sigma_{munu}gamma_5psi$, $bar psi gamma_mu gamma_5 psi$, and $bar psi gamma_5 psi$ in QCD (diagonal quark current-mass matrix). It should be the following:

$$partial^nu langle bar psi (0) sigma_{munu} gamma_5 psi(x)rangle=-partial_mu langle bar psi (0) igamma_5 psi(x)rangle+mlanglebarpsi(0)gamma_mugamma_5 psi(x)rangle tag{1}$$

How do I derive this? This is different than the normal chiral Ward identity, which is:

$$partial^nulanglebarpsi(x)gamma_nugamma_5psi(x)rangle=2mlanglebarpsi(x)gamma_5psi(x)rangle-underset{textrm{anomaly}}{underbrace{frac{N_f}{8pi^2}langle F(x) tilde F (x)rangle}}tag{2}$$

I understand how to derive equation (2) – simply apply a (diagonal) chiral flavor transformation to the generating functional of QCD, then expand to first order (and take into account the anomaly, for example using the Fujikawa method). However I don’t see how to do that for equation (1). The LHS of (2) is $partial^mu J_mu^5(x)$ where $J_mu^5(x)$ is the Noether current associated with the diagonal chiral flavor transformation. However on the LHS of (1), the divergence involves $sigma_{munu}$ which is not the Noether current of any global symmetry.

Also, equation (2) involves field operators evaluated at a single point $x$, whereas equation (1) is point-split.