I admit that I've learned about the KL divergence just now and through the wiki-link, and that my math in general is not so profound. But as it's not about calculation but about the reasoning behind the calculation, I suppose I can have my word:

The wiki-entry mentions that

Typically P represents the "true" distribution of data, observations, or a precise calculated theoretical distribution. The measure Q typically represents a theory, model, description, or approximation of P.

So P here is 10^-18 and Q is either 0 or 0.5.

What your epistemic rationalist has done seems like falling pray to the bias of anchoring and adjusting. The use of mathematical equations just makes the anchoring mistake look more fomal; it's not less wrong in any way. So while the instrumental rationalist might have a reason to choose the arbitary figure of 1/2 (it makes his decisions be more simple, for example) the epistemic rationalist does not. If the epistemic rationalist is shown the two figures of 0 and 1/2 and is asked what approximation is "better" he would probably say 0. And that's for several reason: First of all, if he is an epistemic rationalist and thus trueseeking, he wouldn't use the KL equation at all. The KL takes something accurate (or true) P and makes it less accurate (or less true) KLD, and that's exactly against what he is seeking - having more accurate and true results. But you tell me he has to choose between either "0" or "1/2". Well, if he has to chooce between one of these numbers he will still not choose to use the KL equation. The wiki mentions that the Q in the equation typically stands for "... a theory, model, description, or approximation of P" while the number "1/2" in your example is none of these but an arbitary number - this equation, then, does not fit the situation. He will use a different mathematical method, let's say, subtraction, and see the absolute value of what difference is smaller, in which case it will be 0's. Also, since 1/2 and 0 are arbitary numbers, an epistemic rationalist would know better than use any of these numbers in any equation, since it will produce a result that is accurate just as if would use any other two arbitary numbers. He would know that he should do his own calculations - ignoring the numbers 0 and 1/2 - and then compare his result to the numbers he is "offered" (0 and 1/2) and choose the closest number to his own calculation. Since he knows that the "true" probability is 10^-18 he will choose the closest number to his result which seems to be 0.

Of course, everything that I said about "1/2" above holds true about "0".

(I'm sorry in advance if my mathematical explentation are unclear or clumsy. If I explain arguments through math badly, then I explain arguments through math in English much worst as I was studying mathematics in a different language)