RoscoeD I successfully calibrated the camera with a borrowed laptop screen,
This is a most interesting observation.
Most of the stuff I had read about this camera has to do with the ASI294MC not being able to calibrate with certain filters, but your observation indicates that it also has to do with the light source for the flat.
(I have a non-cooled ASI294MC, but have never needed to calibrate it since I am only using it as an All Sky camera. So this is the first time I am trying to think about the problem -- its has always been "other people's problem" before :-).)
ZWO's data sheet unfortunately does not show any useful QE curves to resolve this if it is light source problem (their curve stops at 700nm), but I managed to dig up an extended QE graph for the Sony IMX294 sensor:

Notice that, like a lot of other Sony back-illuminated Starvis sensor that are meant for security camera (needs good night capability), the IMX294 is also very sensitive to near-IR. (The fact that the response of the blue sensors are still rising at 800nm is also scary.)
The culprit may be the fact that the glass window that is used by ZWO actually leaks IR; and depending on how your optics behave in the near-IR and IR regions can perhaps be causing the resultant flat to not give the correct behavior, especially if the light source is warm.
To maintain proper color balance, color cameras usually come with IR-cut windows. However, they don't need to be very strong, since the non-Starvis sensors lose QE above 700nm anyway).
(See https://www.sony-semicon.co.jp/products/common/pdf/IMX294CJK_Flyer.pdf .)
Given that, my suspicion is that the cause of your problem (and why one specific screen worked for you) could be caused by the camera window not sufficiently cutting off IR.
As such, may I offer one thing to try: if you have an IR-cut filter from a reputable manufacturer, add that to the optical path even though ZWO says the ASI294MC already has an IR cut window. Don't use a ZWO IR-cut filter, which could use the same formulation as what the IR-Cut window.
I know that I have to add an IR-cut filter (I use the Baader Semi-APO filter) with an ASI174MM mini that I point directly at an iPad screen when I use the pair as part of my mount simulator setup. Without that filter, I can see irregular warm pattern on the iPad display.
You will need to take your lights with the same filter that you take your flats with, of course, but if there is IR leak, your lights will also have a better color balance with the IR filter too (kills two birds with one stone).
The one thing that gives me some doubt is why a dark flat (or flat dark) appears to help, and also why longer exposures also tend to help, neither is related to IR. So, don't go out and buy an IR-cut filter just to try the above.
Perhaps @WalterT can do a check since he might be able to add an IR-cut filter to his optical train. Walter -- even an ASI294MM will have this extended QE into the near-IR region.
Anyway, just something I am throwing out that you might try before throwing the camera in the trash, since ZWO still does not appear to be willing to take the camera back.
Chen