I just finished writing up several paragraphs worth of explanation, only for it to all disappear upon hitting "post reply" so I'm going to condense things down in this response...
Basically, CRI does not apply to anything other than "white" LEDs. If a manufacturer is telling you otherwise, they are either mistaken, or misleading you. (unless looking at a freshwater spectrum, but that's a different story, and another show)
CRI when it applies to color rendition does make a very large difference in how colors are represented, with lower CRI not looking as vibrant. Warmer colors in particular don't show up as well under a lower CRI white. You'll see that higher CRI whites also tend to have a more even specrum output as well, which can reduce the need for separate colored LEDs mixed in as much. That having been said, white is not the largest output on most fixtures, that is usually blue LED territory, with white and the other colors filling in the rest of the spectrum. Warmer whites tend to be easier to find in a high CRI than cooler whites as well, so you see a bunch of very high quality whites in the 4000-5600k range with CRI of 90 or above, compared to the 70CRI that is typical of many "cool" white based fixtures. Of course, CRI means nothing to the overall output of the fixture once you turn up the blue really.
In my experience/opinion, a 3000-4000k or thereabouts high-CRI white is what your fixture should be using.