With my ASI294, I believe I'm best taking Lights, Darks, Flats and Dark Flats.
I can't answer your question about integration, but I will offer my opinion on taking darks for flats: I don't think it's worthwhile with ZWO CMOS cameras, which have very low noise. I have an ASI1600MC Pro, and take dusk flats with ACP automation software. My master flats typically show an average of about 0.24 in PI's readout. By contrast, my 600-second master dark shows about 0.012. That's only 5% of the flat level, and would be less if I increased the flat exposure for a higher level.
If I subtract a 0.012 dark from a 0.240 flat, I get 0.228 -- not much difference at all.
But there's another issue: The dark frames should be exposed for the same time as the flat light frames. ACP automatically adjusts each flat's exposure to achieve a target ADU value, so I'd have to manually expose a set of darks dark for each individual flat. It would be a nightmare trying to create a master flat.
Given the wide disparity between my flat ADU and dark ADU, I see no benefit from subtracting darks from flats.