Update to this old thread . . .
As stated above, I have had an issue for a long time in shooting in a studio with monitors on the set. Expensive professional monitors have good color temperature control. Instead, we now have a nice, big, consumer Samsung 4k LED TV's. Their color temp control has "Standard, Cool, Warm1, and Warm2". - which are roughly equivalent (by appearance) to 10000, 9000, 6800, and 5700 Kelvin. Or somewhere in there, it's a guesstimate. Of course my studio lights are at 3200k, so even the warmest of these look bluish on camera.
The monitor needs a SDI to HDMI converter anyway, so I finally broke down and bought a BlackmagicDesign SDI to HDMI MiniTeranex and used the LUT feature to adjust the color temp on the monitor. Looks amazing. I am very happy with the result. What made it especially nice and easy, was finding some free LUTs** for simple color temp changes of this very nature. I am very happy with the color on camera.
The only complaint I have is that the MiniTeranex has a pretty loud fan, which is audible in the studio. A shame. Otherwise this has been a perfect solution. I'm still using it, but do wish it was more quiet, and have to watch my mic placement or make sure it is turned off when doing thing on-set that don't use the monitor.
I love it anyway, but damn.
**Here is a link to the LUTs i used for various color temp/white balances that I found. They were intended for when you have your camera set for tungsten at 3200k, and forget to white balance when you go outside under 5600k or the opposite where you white balance outside and go inside. They have 20 white balance correction LUTs with these white balance values 2800, 3200, 4300, 5500 and 6500 Kelvin allowing you to cross-correct just about any lighting-white balance error you may encounter. . . . . . And they work very well for inexpensive monitors on set, like in my situation.
https://lutify.me/free-white-balance-correction-luts-for-everyone/