I don't think so. Your filter is not "neutral" per say, it plays an active role: it lets more red light in (for ionized hydrogen) and blocks others (it is a mix of broad band pass and and some narrow band blocking.) Your light inside your dome and the reflection off your screen do not recreate a valid sky light and this is what the filter is designed for.
A flat is designed to correct optical defects due to dust in the FOV (camera, mirrors, eyepiece, ...)
Your idea of using a flat to correct a color balance of your CCD chip assumes that you are using a light that emits the same intensities in the RGB and then that your screen will reflect those with the same ratios.
So even without the sky filter, I think that using a flat to get an idea of your camera color bias will give you a "first order" idea.
Retake the flats without the filter, get the color correction and apply them to "normal" pictures (street, people, nature, ...) and see if you can truly get a white-balance correction that way.
You can also take these pictures while placing a grey square in the field of view and see what color corrections you need to apply to the grey square.
