One of the pleasant things about shooting in RAW is that white balance has not yet been applied to the image. The camera will store what the white balance was set to at the time the picture was taken to the EXIF data. RAW editors will default to that value, but it can be changed to any value desired. If a neutral gray or white area is present in the source of the image, that can be used as the white point.
Finding a neutral gray is a little trickier with close up and in focus images taken from a CRT, since there’s really no white. Only the individual phosphors are visible. One way to go about it is to take a photo from a known gray or white screen, like test patterns from the 240p test suite, and ensure the camera is out of focus so no phosphors, moire or other lines are visible. Use that in the RAW editor as your white point for other pictures taken with the same camera and display settings.
I don’t mind setting my camera to 5000K if that is more useful for your purposes, however.