I have found that when I make coconut milk soaps, my whites are a whole lot whiter, regardless of the oils I use & regardless of whether or not I freeze it first. I simply put my coconut milk in the fridge overnight when I'm measuring out my oils & butters the night before.
That being said, if you haven't worked with milks before, they may freak you out a bit as they can get quite ugly when the lye hits it. My last batch looked grainy as all @!#$ no matter how much I stirred my lye in, I was convinced the lye had not properly dissolved (it was the milk fats chunking up actually) and it separated with a layer of yellowish looking liquid at the bottom. Not sexy
Nonetheless, since it was a smaller test batch for some new combinations of natural colorants & not a batch for sale, I went ahead I proceeded as usual. Everything turned out fine. The white portions of the bars turned out very white.
I often use kaolin but don't find it gives a very stark white unless I add so much that it speeds up trace significantly due to the amount of water it sucks up.