The main story is set in modern times, but there might occasionally be glimpses of the past. It's set in a vaguely Lovecraftian (though probably more accurately Derlethian) universe. Without giving too much away, some ancient gods and other nasties are about to break out of an ancient prison, and this society has been anticipating this and trying to develop ways to defeat these entities. While the main defense will be three magical girls, the society is developing various tech, including guns, bombs, and mecha, to try to fight these things as well. This R&D would have to be done through various connected organizations for ostensibly legit purposes (military use, a new type of sport, etc.) in order to maintain secrecy. This use of science as a way to kill demons is where the religion/science mesh comes in.
Don't know if you still need some help with this, but here goes.
It's interesting how much our ideas about the relationship between God and science have changed over the centuries. The Greek word "philosophy" once meant "love of wisdom," and the concept of wisdom in the ancient world almost always included both God and science (with the exception of a few early atheists, who were definitely not the norm!). So "philosophy" in its original sense is a term you could use. Aristotle was a philosopher in this sense -- a man who was trying to unify religious and scientific thought as "wisdom" (or "metaphysics," a term coined by later writers in reference to some of Aristotle's writings.) Aristotle heavily influenced Thomas of Aquinas and the Scholastic theologians of the Middle Ages.
Another good example of such a philosopher is Philo, who is less well known today than Aristotle, but who wrote scads of material in the 1st century CE.
Another term you could look at is "natural theology," which has also come and gone more than once over the centuries. Natural theology tries to give preeminence to reason and science in contrast to pure revelation. There have been a number of offshoots from this line of thinking (eg. Deism).
The term "Gnosticism" was mentioned above, but actually this term would be a very poor choice for the idea of "God and science" blending together, since Gnosticism is a line of speculative thought based solely on "revelation," not science. (Gnosticism does believe in evil entities, however, so from that point of view it could be useful for your story.)
Finally, just to keep all the terms clear . . . the scenario you're describing here (with a secret society devoted to finding ways to kill demons) would most properly fall under the category of
apocalypticism, which has a long and ancient pedigree. (The book called
Revelation in the New Testament is a famous example of ancient apocalyptic literature.)
People these days tend to think of apocalyptic literature as being a form of pure, imaginative, apologetic "revelation," but when you look at the material closely, it's an expression of an underlying faith in the laws that govern the universe. Although many of us no longer agree with this good-versus-evil paradigm to explain the laws of cause and effect in the universe, nonetheless, to the apocalyptic visionary's eyes, it's all about the relationship of God to natural science -- and getting ready for "the big day" when God finally invokes all those nifty "final effect" clauses.
Hope this helps. P.S. In case you couldn't tell, I do academic research in this area and have a degree in theological studies.
Jen