What the New Atheists call the Courtier's Reply concerns the fact that the New Atheist attack on theistic belief is often made in ignorance of what the theist believes. The rebuttal to the reply is that atheists claim that God does not exist, and therefore detailed accounts of exactly what Christians or other theists believe about their God is irrelevant.
In the emperor case, then king is naked, and this can be discovered by looking, and what color the imaginary clothes are supposed to be is irrelevant.
The legitimacy of this response depends on what aspects of theism are relevant to the arguments Dawkins and company are making. It is a matter of what is relevant to the reasons for rejecting Christianity or theism. Admittedly, a lack of knowledge about the difference between Arianism and orthodox trinitarianism is probably not relevant. However, to make this kind of claim, one needs to know what sorts of arguments for theism have been advanced. For example, if you go around saying that you can refute any first cause argument by asking the question who made God, you have to take into account the fact that the causal principles defenders of cosmological arguments use normally don't require that anything and everything needs a cause. In the case of the Kalam Cosmological Argument, the principle is "Whatever begins to exist, must have a cause of its existence." In the case of the Thomistic argument from contingency, there is a distinction between contingent and necessary beings, and while contingent things need causes, necessary beings do not. So your defense of the claim that all cosmological arguments fail in this way will inevitable come across as ignorant to people who know something about how such arguments are supposed to work
If your claim is that the God of the Bible is morally deficient, then you have to have some understanding about how Scripture passages are interpreted and understood by theologically informed religious believers. You could make the case against God without bringing any of this up, but if it is part of your case against God, then you need to do your homework and understand what believers actually say about this.
If your argument is that religion conflicts with science, then you have to take seriously the kinds of attempts that are made to reconcile religion and science by people who have considered the question.
Hence, some aspects of theology are going to be relevant to the arguments you would be making, and others may not be.