After shooting around the house for awhile, I'd like to say that the AWB on the 1DX is actually really, really good.
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These shots with indoor and mixed lighting where I can't use a flash are difficult. I think WB does matter a great deal, especially if it adds a cast to a shadow that you can't get rid of. I don't think you can ignore that. Fortunately I was able to test both the 1DX and 5D3 in these harsh conditions and they performed similarly.
You know, you can always develop the RAW twice, once for each white balance, and then composite the one into the other using a mask and a big soft brush....
For white balance, your first best option is to use a styrofoam coffee cup as a white balance target. Styrofoam has a nearly perfectly flat spectral response, far superior to that of any of the commercial white balance targets. It's also about 80% reflective, which is excellent for setting white balance; low enough that there's no danger of clipping (unless the photo is hopelessly overexposed), yet high enough that noise isn't a factor. You can set it in the scene and get a nice sampling of each of the light sources by direction (say, daylight coming from the right and fluorescent coming from the left, blended roughly equally in the middle); or, you can put it over the lens and get a good whole-scene sampling suitable for an in-camera custom white balance.
If you don't have a spectrally-flat target (and note that many common "white" objects such as shirts and paper and the like are far from white), your next best bet is to boost the saturation in post-processing to 100%, fiddle with the white balance until it looks as neutral and un-saturated as possible, and then return the saturation to wherever you want it for that picture. It's much easier to tell when the white balance is off when the picture is over-saturated.