How would you handle the integration of a non-atheist concept in a sci-fi universe, while still making it feel like sci-fi and not Fantasy?

1. Look at the V-A-S-T body of philosophical thought that deal with that. The idea that nature proves god used to be really common and even today far from disappeared. Many scientists believed they Knew it existed. I know a pretty devout Catholic microbiologist who works on creating a cure to aging for example. So just treat those arguments as valid.
2. The key difference between magic and miracles is that the latter is considered exceptional. So 99.99% of the time things are gonna proceed according to rather predicable laws.

Do you know any non-atheist sci-fi stories?

Hmmm. Perry Rhodan arguably counts. Kinda.

To which then I wondered "How would a sci-fi story deal with having a demonstrably existing god in its universe?". And I'm not talking about having religions, whose beliefs are treated in the story as mere possiblities at best, or a secular cult a la bolshevik God-Building. I'm talking about integrating a god (or any non-atheist concept for that matter, like deism or pantheism) into the workings of that universe, and the knowledge and societies of the people living in it.

Could be interesting.

In my current project there's not much direct evidence of god but I've realized that with things being as they are the overall makeup of the various civilizations would likely trend towards the devout in most cases, so I get where you're coming from.