I understand your concern, and I am not saying what you think I'm saying, so allow me to clarify further.
A rational basis for a belief is a basis that cannot be denied by a rational adult, i.e. this is why a rational adult should believe this. A rational basis is similar to a "proof" except that it involves all science, math, logic, economics, observations, empirical evidence, etc... It does not mean that a belief that does not have a rational basis should not be believed, it only means that it could/might be denied by a rational adult. One could preach a particular religion, e.g. Christianity, and it might very well resonate with some and they might believe it. The key is that some rational adults might reject the faith as well. Any proselytizer of the faith would not be able to call those who reject the faith irrational solely on the basis of the rejection.
On the other hand, if you were to go on a vacation for two weeks and not arrange for anyone to feed your fish because you refuse to accept all explanations for how they will be dead upon your return, you could be labeled as irrational (on that matter). There is a compelling reason for you to hold a particular belief that is founded in science/math/logic/economics/etc, and you could be labelled irrational solely on the basis of the rejection ... unless you actually want your fish dead.
That is all that means. A rational basis means that a rational adult must accept. An irrational belief may be denied by a rational adult (or may be accepted). Religions are inherently irrational (per this definition) and may be denied by a rational adult.
Okay...you are trying to redefine 'rational'.
This word first appeared in the English lexicon around the 14th century, stemming from the Latin rationalis, which means 'of or belonging to reason'. A 'reason' only need be logical. Since ?A->A is a valid logical expression, the circular argument itself is logical. All religions are based on some initial circular argument, or argument of faith, with arguments stemming from that. In other words, the circular argument itself (along with any supporting evidence) can be reasonable in and of itself.
Having said that, the Church of Global Warming is NOT reasonable, since it blatantly ignores various theories of science. The circular argument itself is at odds with these theories. That circular argument is not a reasonable one.
Christianity, for example, is reasonable, but not provable. Because it's reasonable, it does not conflict with any logic or any theory of science or any mathematics. Logic is still satisfied with ?A->A.
You cannot use logic or mathematics or any theory of science to prove that God does not exist. You cannot use logic or mathematics or any theory of science to prove that Jesus Christ does not exist.
The same can be said for Shintoism, Buddhism, even some forms of paganism. It can even be said for the Church of No God.
Atheism, is of course, different. Being the only non-religion, it needs no circular argument or argument of faith. Because you follow this type of reasoning, you tend to ignore that ?A->A is a valid logical expression. It stems directly from the Law of Identity, an axiom in logic.
You CAN use theories of science to show the circular argument defining the Church of Global Warming to be itself irrational.
You CAN use mathematics to show some of it's supporting arguments are themselves irrational.
To group ALL religions as 'irrational' is a compositional error fallacy.