Yes, they are related. If a light like a white LED is slightly deficient in red, giving it a lower CRI rating, white balance will make the camera boost the red in relation to blue and green. Cameras only care about the ratios between red, blue, and green. As long as you don't have a royal blue or a violet light source, a decent camera should be able to white balance a slightly off-white source into what looks like neutral white.
Like I said, that may be OK for home video or ENG, but not for studio lighting:
(1) If you boost the red by using white balance on your video camera you introduce lots of grain into the red channel.
(2) There's a discontinuous spectrum problem too. Some wavelengths are completely absent from the output of the light, therefore you can boost all you want but they're not coming back.
(3) You assume that the LED light is the only light. If there are other light sources and you change white balance, well then you won't get a consistent color unless you gel the other light sources to match, which might be difficult if the other light source is, for example, the sun, or the sun reflecting off some large surface.
(4) Gel the LED all you want, you can't fill in the gaps in the spectrum.
(5) Add some red and cyan/green LEDs to balance things out and now you see the weird extra colors whenever the light reflects on any shiny surface or makes a hard shadow. So you diffuse it to get rid of that, well now you can't use it undiffused and you lost a lot of output.
(6) There are other cameras besides video cameras out there. Some of them still even use actual emulsion. And those don't have white balance when I last checked.
(7) Do you really want to have a $300 an hour plus color correction process dealing with issues like this?
(8) The P7 wasn't designed for this application. So why use it when there are more appropriate alternatives available? If you're spending hundreds or thousands of dollars on LEDs is it that hard to pick up the phone and call a manufacturer in order to place an order?