Well, you could try desaturating your flats, but honestly I wouldn’t bother. It’s perfectly normal for images to need color adjustment regardless of whether you calibrate or not. Local light pollution, variations in sensitivity in the different color bands, variations in transmission in dyes in your CFA, and lack of a white balance “feature” in dedicated astronomy cameras mean you will have to adjust color on deep sky images almost no matter what.
The specific process for addressing this varies from one software package to another, but the basic process is the same. A pedestal is added or subtracted from each color channel to get a neutral black point, then a multiplier is applied to each channel to address differences in sensitivity. Finally, a color gradient may need to be applied to make sure color cast is even across the frame. In PixInsight, for example, that is performed with a combination of dynamic background extractor, background neutralization, and color calibration. In Photoshop one could use a gradient tool (if the gradient isn’t too complex), and a pair of levels adjustments or a levels adjustment with a curves adjustment.
I would not try to address the color cast introduced by your flats. It honestly won’t hurt anything. In your example, the uncalibrated version has a slight green cast while the calibrated version has a stronger red and magenta cast. Either way you’ll want to correct the colors.