Trying to understand the Chiral anomaly, I decided to explore the simplest example of a holomorphic fermion in 2D in a background electromagnetic field $A\text{d}z+\bar{A}\text{d}\bar{z}$. The Euclidean action looks like $$S(\bar{\psi},\psi)=\int d^2z\,\bar{\psi}\bar{D}\psi,$$ with $\bar{D}=\bar{\partial}+\bar{A}.$
The classical symmetry $\psi\mapsto e^{i\epsilon}\psi$ and $\bar{\psi}\mapsto e^{-i\epsilon}\bar{\psi}$ should be anomalous. In order to check this, I wanted to explore the renormalization of the conserved current $j=\bar{\psi}\psi$. In the literature I've seen the difficulties associated to using Pauli-Villars or dimensional regularization, so that I wanted to try something different. I decided to use heat kernel regularization, so that I replaced the $\bar{A}=0$ propagator $$\langle\bar{\psi}(z)\psi(0)\rangle=\frac{1}{z}=\int_0^\infty\frac{\text{d}t}{t^2}e^{-|z|^2/t}\bar{z}\mapsto\int_\epsilon^\infty\frac{\text{d}t}{t^2}e^{-|z|^2/t}\bar{z}.$$
Of course, since we are treating the electromagnetic field as a background field, the theory is free. Thus, renormalizing $j$ simply corresponds to normal ordering it, i.e. removing the divergent part of the loop diagram obtained by contracting the two fermions in $j$. However, since $\bar{A}$ spoils translation invariance, we can instead treat the $\bar{\psi}\bar{A}\psi$ term as an interaction, transforming the single loop diagram of the full theory to a sum of single loop diagrams with the $\bar{A}=0$ propagator and photons coming out of the loop. The number of photons in the loop is the power of $\bar{A}$ in the final result.
Since $[j]=[A]=1$, only the diagram with one photon coming out is potentially divergent (the diagram with zero photons is equal to zero since in this regularization $\langle\bar{\psi}(0)\psi(0)\rangle=0$) and the divergence should be logarithmic $\ln(\epsilon/\mu)$, with $\mu$ some subtraction scale. However, when I compute the diagram I obtain $$-\int_\epsilon^\infty\frac{\text{d}t_1}{t^2_1}\int_\epsilon^\infty\frac{\text{d}t_2}{t_2^2}\int\text{d}^2z\,e^{-|z|^2\left(\frac{1}{t_1}+\frac{1}{t_2}\right)}\bar{z}^2\bar{A}(z).$$ As we explained below, the divergence can only be proportional to $\bar{A}$, not its derivatives. Thus, expanding $\bar{A}$ in a Taylor series, we need only keep its constant part $\bar{A}(z)\mapsto \bar{A}(0).$ The remaining integral is however a Gaussian integral that vanishes. Indeed, $$\bar{z}^2=x^2-y^2-2ixy.$$ The $xy$ term won't contribute since there is no propagator for it. And then the $x^2$ and the $y^2$ contributions will vanish due to rotational invariance.
It seems like $j$ receives no quantum corrections and thus I don't see how it can become anomalous. Furthermore, even if I made a mistake in here, even if $j\propto\bar{A}$, the conservation law is going to become $\bar{\partial}j\propto\bar{\partial}\bar{A}$, Once the antiholomorphic moving fermion is added, the axial current would be obtained by subtracting the two currents together. This would leave us with an anomalous term proportional to $\bar{\partial}\bar{A}-\partial A$, instead of proportional to $F=\partial\bar{A}-\bar{\partial}A$. Furthermore, I don't see how the anomaly would cancel for the sum of the two currents, so that the vector symmetry can be gauged. I feel like there is a conceptual point that I am missing about how these anomalies work.
