But quite a few atheists say that claims about anything that is outside science-in-the-second-sense's domain is irrational. Example: Alex Rosenberg states that any knowledge outside of physics' domain is irrational (in line with his reductionist philosophy, he holds chemistry, biology, etc to be physics on a bigger scale). Hume also seemed to be pushing such views with his fork. This claim is trivially easy to refute, I assume you know the arguments already so I won't waste your time with them.
I'll even go further and claim that distinction itself is meaningless and only matters when say organizing university departments.
What needs to be kept in mind is emergence. Chemistry emerges from physics, then biology from chemistry, ecology from biology, etc. Experimentally verified God will become the base level out of which physics emerges from in the other direction. In that sense, physics' domain perhaps includes everything. But it all depends on how we define the terms.
> the purpose of the intellect is the attainment of truth
This I wholeheartedly agree. Maybe you'll enjoy reading this blog post [0] that proposes Truthism as life's goal.
[0] https://waitbutwhy.com/2014/10/religion-for-the-nonreligious...
For instance, if the 5 proofs had actually logically and definitively proved that a personal God exists, then yes, that knowledge would become part of science.
Conversely, is it not true that the actual bedrock of religions are the books/revelations. Imagine a world where were no such revelations. Would people have as much faith in a personal caring God just on the basis of proofs?
And if the revelations were actually shown to be true, that would become part of scientific knowledge. Say God appears on Earth today and agrees to undergo scientific observations of His nature (say by turning water to wine or parting the ocean under experimental scrutiny), then naturally His existence will have to be accepted as part of reality.
The scientific method is used by everyone, whether consciously or unconsciously and to the best of their ability, to survive and understand the reality we observe. There is no other way to judge right from wrong. Scientists obviously use it to study physical reality. And theists use it for instance to judge which among the various religions (and which denomination within a religion) is actually true. Yes, that is also an application of the scientific method, looking at the arguments for the different religions and judging which one (or none) seems to be true. The disagreements come in when we don't have enough data to make a definitive judgement between the alternatives. And that's where all the wonderful imaginative ideas continue to exist.