Nielsen-Ninomiya Theorem versus Chiral Gauge Anomaly

I think the Nielsen-Ninomiya theorem is more closely related to a gravitational anomaly than a $U(1)$ anomaly. For instance in 1+1d, two left-moving fermions of charge +3 and +4 and a single right-moving fermion of charge +5 has a vanishing $U(1)$ anomaly, since $3^2+4^2 =5^2$. However, it still cannot be put on a lattice, because it has a chiral central charge.

Here is a recent paper which addresses why a 1+1d chiral theory (in the central charge sense) cannot be put on a lattice, from the perspective of energy currents: https://arxiv.org/abs/1904.05491 . The up-shot is that a local lattice model cannot have a nonvanishing energy current in the ground state, which is exactly what a chiral CFT has.