I have my own code for aligning and stacking and the asi1600 seems prone to grid effects like these. I'm not sure why.
One way this can happen - as a registration artifact - is by making parts of the image "more smoothed" than others. In that case it isn't a matter of a bright grid with dark squares in it - but a noisy grid with more smoothed squares in it - and the parts with more noise - and higher standard deviation - just appear brighter to the eye. But what's really happening is the stack ends up with a pattern with varying noise.
If that is the case there is no way to flatten it by subtracting or dividing out a flat. The mean value across the image is correct - but the noise varies.
The asi1600 may be more prone to it because it has low read noise or something. I'm not sure.
I don't think the flats play a role because flats mainly impact the bright parts of an image - but this artifact happens in the dark parts.
Other stacking methods may show the effect less - but I would make sure they aren't just smoothing the image more under the covers - which is undesirable if you want max detail.