The whole image is bluish; and the shadow is bluish everywhere, not only behind the cover. I would start with setting the correct white balance with a grey card
Ok. For now, forget the gray card, you already have two places where to extract the "gray-white" balance from. The background itself and the source light.
If your camera supports custom white balance, take a photo of the background, not overblown or overexposed, but underexpose it until it looks grayish. Then use the image as white balance.
There are cons and pros about using the background as the white balance target.
The pro is that the background will look white... This will be the definition of white itself on your image.
The Con is that if it is the case that the background is not actually white, but, let's say a slightly bluish or whatever, all the colors will shift a bit to the opposite side, a bit warmer in that case.
If I chose then, to use the light itself as a white balance target, I would need to mask the background to correct it a bit... which implies more work; more accurate colors, but more work, so, for now, use the background.
If the shadow is blue because the case is blue, I would leave it as such.
I would expect a good enough CRI on the light your box has, but regardless of the color accuracy of the light, the white balance can be almost fully controlled targeting the background or the light source.
What would be the difference between fixing the white balance in the shot, vs in post?
Normally, both would work, but in your case, you are blowing a lot the white, so this renders the white unusable to fully fix the white balance in post because you are clipping color information. (Every color overexposed enough turns white.)
This is why you can half fix your current photo, yet have some blue in the shadows.