First consider how the moral realist could respond if the targeted "believers" (whose reliability was being put in question) were not themselves, but some alien species. In that case, we may respond to Street by acknowledging that, in the absence of further info, we may have no antecedent reason for expecting this alien species to be morally reliable. But rather than trying to assess their reliability more or less a priori, from only very abstract info, we would do better to look at their concrete moral judgments and practices. If we look and find that the aliens seem generally sympathetic and altruistic, concerned to promote the wellbeing and non-harmful life goals of other sentient beings, then we may be reasonably confident that they're on the (morally) right track, whatever the causal process that brought this about. (If, on the other hand, they seem to enjoy gratuitous torture, then we'll judge them to be morally abhorrent.)
Of course, using our first-order moral judgments in this way is no longer going to be dialectically effective when it is our own reliability that is in question. But that shouldn't necessarily concern us (see also my Railtonian defense of 'default trust'). As Parfit writes in On What Matters:
[Street holds that] we must show that the evolutionary forces have led us to form true normative beliefs, and we must defend this claim without making any assumptions about which normative beliefs are true. What Street here requires us to do is impossible. Some whimsical despot might require us to show that some clock is telling the correct time, without making any assumptions about the correct time. Though we couldn’t meet this requirement, that wouldn’t show that this clock is not telling the correct time. In the same way, we couldn’t possibly show that natural selection had led us to form some true normative beliefs without making any assumptions about which normative beliefs are true. This fact does not count against the view that these normative beliefs are true.
I think this is exactly right. That's not to say that our first-order normative beliefs necessarily suffice as a decisive refutation of Street's skeptical argument against realism (there's more to Street's argument than there is to the whimsical despot's demand), but nor can they be dismissed as entirely irrelevant or 'illegitimate' to appeal to. If we want to work out what to believe, we need to consider all our beliefs -- everything we find most plausible, and judge to be (likely) true -- in wide reflective equilibrium, and work out how best to fit the various claims together: what to keep, and what to discard, to yield the overall most plausible conclusions.
At the heart of Street's argument is: (1) the observation that realists must consider themselves lucky (isn't it a wonderful coincidence that natural causes just so happened to influence us to hold moral beliefs that roughly align with the mind-independent moral truth!?), and (2) an epistemic principle to the effect that there's something epistemically irresponsible about retaining one's belief in the face of such luck.
I agree with the first premise: as realists, we must consider ourselves epistemically lucky. And the second premise has a great deal of prima facie plausibility to it. But it's not nearly as incontrovertible as Street seems to assume. For one thing, I certainly don't think that plausible-seeming abstract principles necessarily trump plausible-seeming concrete judgments, e.g. that gratuitous torture is wrong regardless of whether the torturer herself is committed to this conclusion. Secondly, there are more direct grounds for doubting the principle, as it often seems completely appropriate to consider ourselves epistemically lucky, e.g. if we narrowly avoided being brainwashed, or raised by a cult of counterinductivists, or some such.
To bolster her case, Street compares the moral realist to someone who stubbornly believes herself to have won the lottery, even before hearing the results of the draw:
At this point, however, the normative realist is in no better position than the person who question-beggingly insists that she won the New York Lottery, even though she has no reason to think so apart from the fact that she entered it. If we are normative realists, we think there is a “winning” coherent system (or systems) of normative thought; we also think there are countlessly many false coherent systems of normative thought, which, but for sheer good fortune on our part, causal forces might have shaped us to endorse; we think that as it so happens, ours is among (or approximating) one of the true ideally coherent systems; but when asked to give our reason for thinking so, all we can say is to repeat, in so many words, that it is among the true ones—to insist that we, and not the countless number of mistaken possible others, “see” or “sense” what is normatively true. But this is no better than insisting, without any non-trivially-question-begging reason to think so, that one has won the New York Lottery. Given the odds we can reasonably suppose to be in play in this “normative lottery” case, we should conclude that in all probability we didn’t win—that, if there is indeed such a thing as the robustly independent normative truth we are positing as a substantive normative premise, then we are probably among the unlucky ones who (just like the ideally coherent Caligula, grass-counter, hand-clasper, and so on) are hopeless at recognizing it.
The main problem with this analogy concerns the "odds" that we assign to the various outcomes. Given our understanding of the lottery mechanism, we know that we should assign equal odds to each possible outcome, and hence the odds we should assign to some particular ticket (#139583923, say) winning is extremely low. For Street's argument to work, it must be that we are similarly required to assign equal (or roughly equal) odds to the truth of each possible normative system. But why think that? I certainly don't assign equal odds to all possible normative systems -- I think it's overwhelmingly more likely that pain is intrinsically bad than that it's intrinsically good -- and I don't see anything in Street's argument that suggests I should change my mind. (It's not as though the normative truth was itself settled by a chance process that gave equal objective probability to each possible outcome.)
Street might try to push the analogy by asking us to imagine a lottery winner who likewise insists that the (a priori) odds of ticket #139583923 winning are much higher than for any other ticket. There are a couple of things to say about this. Firstly, I agree that such a person would be crazy. But it's far from clear that there's necessarily anything incoherent about their beliefs: they might just have wacky priors, like the epistemic counterinductivist. I think they're being irrational as a matter of mind-independent normative fact, but of course Street cannot think that. What's more, my reasons for thinking that equal odds are required in the lottery case (viz. something like Lewis' "Principal Principle" relating objective chances to rational credences) simply don't carry over to the normative case, where the truth of the matter was not settled by a chance process.
So I think Street's argument strictly fails: There's nothing incoherent about assigning higher odds (a priori) to anti-pain normative systems than to pro-pain views. So the moral realist needn't conclude that she is "probably" among the misguided when she holds to her anti-pain view. It's true that our views are formally analogous to others (like the pro-pain view) that we hold to be mistaken. But I don't think there's anything necessarily wrong with the form of those other views -- they're just wrong on the substance. So I'm not particularly troubled by sharing the form of the wrong-headed view, just with different substance. That's how the right view would have to look.
Suppose you grant that the realist needn't (on pain of incoherence) take herself to be "probably" among the misguided. Still, you may think, perhaps it's still the case that she really should conclude this. But now you're positing a mind-independent normative truth. And if the argument against normative realism depends upon the truth of realism, then that can't possibly be a sound argument!
So there's no sound argument against realism to be found here. But there is at least a challenge for the realist, to say more about how moral knowledge is possible on their view. I can't give a full answer here, but here's a rough sketch of a view: There's a fact of the matter as to which psychologies qualify as fitting or 'substantively rational'. (The fitting psychology might, among other things, endorse modus ponens as a rule of inference, accept inductive over counter-inductive norms, and take the badness of pain as a provisional moral datum. The rough idea being that the 'fitting' psychology is one that reflects or fits with the normative facts.) Inferences drawn, and conclusions reached, by agents with fitting psychologies are ipso facto justified -- and if true, eligible to qualify as knowledge.
That seems a coherent view, and one that allows for (fitting) agents to know the mind-independent moral facts. One might think of it as similar to a kind of reliabilism (the view that true beliefs formed via reliable methods thereby constitute knowledge). Reliabilism is clearly immune to Street-style skeptical arguments: the mere fact that there are other, less reliable, processes does nothing at all to undermine the processes that I actually use (even if I have no independent way of verifying that the process I use is one of the few reliable ones). And it brings out the disanalogy with the lottery case: presumably there's no way to be a reliable predictor of lottery results, the way that an anti-pain moralist might be a reliable predictor of moral truths. Similar observations may be made when using the fittingness view in place of reliabilism.
In short: it's epistemically undermining if this psychology I'm using was 'lucky' to get things right. That casts doubt on the rationality of my psychological processes. There's no such problem if I'm merely lucky to have acquired a reliable psychology in the first place. That doesn't cast doubt on the rationality of the psychology I have; it merely suggests that I could easily have ended up with some other, less rational, psychology instead. To which the appropriate response is just, "Well thank goodness I didn't!"