Is it true that natural peanut butter splits in cookies?
I've made peanut butter cookies with various "all natural" peanut butters, containing no extra oil/fat, just peanuts and possibly salt. They didn't split. I suppose the recipe you're looking at could be somehow different but it seems really unlikely. I haven't even seen splitting in cooked sauces using these kinds of peanut butter, along with plenty of other liquid.

As far as I know, the main thing meant by "all natural" is that it doesn't contain other oils like yours does; those are generally used to replace some of the peanut oil, so that it doesn't separate while in the jar. But separating over a long period of time in the jar is a far cry from splitting in cookie dough where it's mixed with plenty of other ingredients, and probably sits around for a few hours at most before baking.