The color play of sun & sky will vary according to haze, clouds, and sun elevation. But often HDRs are adjusted to be more neutral to white overall. However, the color distance between the sun and sky should be correct.
With that in mind, I think the correct procedure is to use a coloring addon (I use Photographer, good enough) and check against real photos set to some neutral’ish (5500K-6500K should be ok) color balance on the camera and set Photographer addons color balance to same. Now adjust the colors of the HDR to be approximately the same in Blender as in the photo.
You now have set a reference. If you now do indoor lighting and WB for 2700-3000K lights (making them neutral white), you adjust Photographer addons white balance and the blue sky outdoor will become even bluer.
This is what the camera and eyes will see. However, your brain plays tricks on you and outdoor and indoor will appear to be much closer together. And it is an effect photographer don’t like at all, making them flash a site with close-to-outdoor WB flashes to balance out the WB (and lighting as well). But we’re lucky - we just lower the effect slider of the HDR coloring RGB Mix node. Here a real photo, just to illustrate how “blue” the outside really is when it’s dimmed down (late morning) and overcast (no sun, so more blue), when the cameras WB is set to indoor. If camera WB was set to overcast, the outdoor would go more toward neutral but the indoor lighting would get extremely yellow.