no, CCT does not change CRI
whether a light looks Cool or Warm, depends on the White Balance of the operators brain at the time.
IF the operators brain is adapted to sunlight, then a CW light will seem white, and a NW light will seem warmer (yellowish).
otoh, IF the operators brain is adapted to darkness (takes 30 minutes in the dark), then the CW will seem blueish and the NW will not seem yellowish.
the following photos are from iPhone 8 with Automatic white balance. Notice how their color changes based on what they are being compared to. Remember the camera is changing its white balance in every shot. IF the photos were taken with a camera with a Manual white balance setting, then all the colors would stay the same. (and what those colors would be, would change based on which white balance Color Temperature was being used)
A light will look blueish if it is cooler than the white balance of the brain at that time. And a light will look yellowish if it is at a lower CCT than the brain is adapted to at that time.
CRI is a separate subject
Tint is a separate subject also
The following 3 LEDs are all the same 90+CRI, but their R9 (RED) values are not the same. R9 is not part of the 90CRI calculation. The 3 LEDs also have different tints. The worm has the most neutral Tint, the S1 Mini has the greenest tint, and the sw 45 219b has the most, what would you call that Tint?
The CCT of each LED is also different, 4000k for the worm, 5500k for the S1 Mini, and 4500k for the sw45
Tint color is plotted above or below the BBL, the black line (most green top left, most magenta bottom right)
Color Temperature of the light plots along the BBL (most orange on right, most blue on left)
CRI is NOT shown on this chart
White balance is along the BBL, can be at any color temperature, and gives rise to the term Neutral Tint, not to be confused with Neutral White (which is a range of Color Temperatures, Warm, Neutral, Cool)