So, I assume that you are using your 1/4000 Bias as a calibration frame for your Flat? And I also assume that you are taking the Bias frame as if it was a Dark, with the lens cap ON?
Personally, I would rather IGNORE the whole concept of Bias frames. They are just NOT necessary if you have the ability to control exposure time (which you do) and CCD temperature (which you can).
Instead of Bias frames (at 1/4000 sec) take FlatDarks (at 1/250s, or the same as your Flats).
Remember that a Flat and a FlatDark both contain (effectively) exactly the SAME 'bias' information and that, when you DarkSubtract the FlatDark from the Flat, you will successfully eliminate ALL traces of the Bias frame altogether (well, that's the theory, anyway).
And, exactly the same goes for your Light and (Light)Dark frames - dark calibration gets rid of the Bias information, as well as removing the Dark signal from your Light.
Does that make sense?
It shouldn't really affect the time taken for you to acquire the data. You will need to take, say, 20 FlatDarks at (say) 1/250" (whereas you might have been taking 20 BiasOffsets at 1/4000"). Overall, this is going to 'cost' you an extra 20 x 1/3750" - and I am sure that you will be able to spare that extra 5mS
Whether this eliminates the 'striping' or not is another matter altogether
Cheers,