From Cheating Death in Damascus (bold emphasis mine):

It’s less clear how we should model this case from the point of view of FDT, and there are a variety of options. The most natural and illustrative, we think, is to assume that what actions you would or would not perform in various (hypothetical or real) circumstances determines whether you’re a psychopath. What actions you would perform in which circumstances is in turn determined by your decision algorithm. On this reading of this case, the potential outputs of your decision algorithm affect both whether you’d press the button and whether you’re a psychopath. 
In practice, this leads the FDT agent always to refrain from pressing the button, but for very different reasons from the CDT agent. [The FDT agent] reasons that if she were to press the button that kills so many people, then she would be a psychopath. She does not regard her psychopathic tendencies—or lack thereof—as a fixed state of the world isolated from what decision she actually makes here. 
This seems like the right reasoning, at least on this understanding of what psychopathy is. Psychopaths just are people who tend to act (or would act) in certain kinds of ways in certain kinds of circumstances. We can take for granted that everyone is either born a psychopath or born a non-psychopath, and that [the FDT agent's] action cannot causally change this condition she was born with. Yet if this condition consists in dispositions to behave in certain ways, then whether [the FDT agent] is a psychopath is subjunctively tied to the decisions she actually makes. If you would not perform $A$ in circumstances $C$, then you also would not be the kind of person who performs actions like $A$ in circumstances like $C$. FDT vindicates just this sort of reasoning, and refrains from pressing the button for the intuitively simple reason of “if I pressed it, I’d be a psychopath” (without any need for complex and laborious ratification procedures). When we intervene on the value of the $FDT(P,G)$ variable, we change not just what you actually do, but also what kind of person you are.

I'd give the psychopath button question a similar answer that I would to the smoking lesion question: what's the mechanism that correlates being a psychopath with willingness to press the button?

If being a psychopath (or not being a psychopath) affects your answer because it affects your ability to reason, then based on your psychopath status, you may not even have the ability to reason correctly and choose an outcome. The problem is ill-defined, because it asks you to do something that you may be incapable, by stipulation, of doing.

If it affects your answer in another manner, then pressing the button because of the outcome of a reasoning process won't be correlated with psychopathy even though pressing the button in general is. (Unless the button uses your decision as a criterion of psychopathy, in which case we get into halting problem considerations.)

Also, note that in everyday language, "only a psychopath would press the button" strongly implies that it affects your decision because it affects your values about killing people, which is the second scenario. It's also inconsistent because the problem statement implies that both psychopaths and non-psychopaths would consider pressing the button and would reject pressing the button only after figuring out the logic, but if a non-psychopath would always refuse because of his values, this implication isn't correct.

(Edit: Edited this lots of times. Phrasing my objection correctly is actually quite hard.)

In my view, FDT handles the problem as follows:

Frank: Suppose FDT(situation) = "push the button". Then all psychopaths die, which includes me. Suppose instead FDT(situation) = "don't push the button". Then no psychopaths die. Since I prefer living in a world with psychopaths to dying, FDT(situation) = "don't push the button".

The main controversial piece is from the problem specification: "Paul is quite confident that only a psychopath would press such a button." I think this mixes up P(button|psychopath) and P(psychopath|button), but since the problem specification is our only source of how the button determines who is or isn't a psychopath, it seems fine to trust it on that point.

Another related problem is one where there's a button who kills everyone who would, given the option, press it. You might expect that such people are bad neighbors and prefer a world without them without having any way to act on that belief (and if you come to believe that FDT pushes that button, what it really means is that you shouldn't be so confident people who would press the button are bad neighbors!).

[In general, your decision theory should save you from claims in the problem specification of the form "and then you make a bad decision", but it can't be expected to save you from having incorrect empirical beliefs.]

Psychopathy is strongly associated with poor impulse control and low self-reflection. If Paul is considering logical decision theories, rational choice, and their possible ramifications given his own mental makeup then he is substantially less likely than baseline to be a psychopath, which generally make up on the order of 1% of the population.

Does he have some prior evidence that he is a psychopath? If not, then his prior should be on the order of 0.2% or so. Willingness to press the button would otherwise be his only evidence, which he is "quite confident" about. What numerical value should he put for "quite confident"? Let's say 90% (much more than that should be described more like "very" or "extremely" confident). So that would bring a baseline prior up to the order of 2%.

Now he "very strongly" prefers living in a world with psychopaths to dying. Is that 5x in utility? 100x? 10,000x? Well, dying is a pretty bad thing but I'd use some stronger term than just "very strongly" for 10,000x so let's go with something on the order of 100x.

Well, this is awkward. For outcomes of pressing the button we've got a credence of 2% for being a psychopath and -100 utility, versus 98% for not being a psychopath and +1 utility. This is a net -1 utility, but the numbers are only order of magnitude estimates so the expected value could easily be much more positive or negative! It doesn't really matter which decision theory he uses, Paul just doesn't have enough information.