This seems like a very strange way to look at it. It is true that the Abassid caliphs were very tolerant of different cultures, approaches, religious interpretations. Their empire in part required it — it was vast, Muslims were often in the minority in many of their cities, and they were rich and successful. It is easier to be tolerant when you are comfortable, and Islamic law gives plenty of allowances for tolerance and dealings with non-believers.

Over the course of the period, something shifted, and it's hard to exactly pin down what, but there are a variety of arguments. The sum total is that the empire came under strain from a variety of sources (Crusades to the West, Mongols to the East, and a loss of some of their economic prosperity as trade routes changed), and one of the responses to that was an increased intolerance for secular ideas, other cultures, etc.

But we shouldn't see fundamentalism or intolerance as the "true" form of any religion. It is just a form of it. Religion can be used to spur scientific work, it can be used to smother it. The same religions can, and have, done both. Islamic respect for book knowledge, for natural theology (understanding the nature of God through his works), and for practical, secular knowledge (i.e. medicine) was definitely a fueling part of their "Golden Age." To posit that somehow the Houses of Wisdom, the translation efforts, the hospitals, etc. were merely a "toleration" is a very backwards and misunderstood interpretation. Islamic rulers can immense material support to institutions of knowledge, and that is why they had their "Golden Age." Eventually that support withered, as it so often does.

As for crediting "Islam" vs. "occupied culture" — you are talking about a throughly mixed empire. Consider what the stakes are in trying to decide whether you want to give credit to one versus another culture, rather than seeing their context as inherently intertwined. Consider the role of the Houses of Wisdom, created by caliphs as observatories, scriptoriums, and translation centers — how do you separate out the religion from the culture, there? They are completely intertwined. It is a fruitless question to ask in this case. One does not get a bounty of scientific output merely by tolerating it. As a point of contrast, one might consider the Romans, who tolerated scientific work but did not really encourage it. There are a few Roman high-points in science — e.g., Galen, who was, by birth, a Greek — but for the most part they were relatively barren compared to the Hellenic and Hellenistic Greeks. The number of important Arabic/Islamic scientists/philosophers is much vaster; it was not a merely "tolerant" culture, but one that promoted learning and knowledge.

As for religion and science — the "religion always is an enemy/obstacle of science" (what is known as the Draper-White "Conflict" thesis) is reliant on extreme cherry-picking and often straight-up misunderstanding. No serious historian of science finds it plausible. What we find if you look at the record is that religion is just one of many cultural and sometimes political forces at work in any given society. Sometimes these forces promote scientific work, sometimes they discourage it. Sometimes they promote some but discourage others. It is a mixed bag. There are many eminent scientists who were explicitly motivated by religious beliefs (Isaac Newton is a famous one), there were many religious institutions that provided a strong context for scientific work (think Gregor Mendel in his monastery). To come up with some kind of blanket rule about the relationship between science and religion is as silly as coming up with a blanket rule about science and politics.

What makes science interesting in both a religious and political sphere is that it can serve as an alternative source of authority and power. (The sociologist Bruno Latour has posited that it is one of the few places where new sources of authority and power get created in a society.) So in that respect it often finds itself either being used by other powers in society (think of the application of science to war), or at times struggling with it (think about how climate scientists have struggled with the fossil fuel industry). Religion is another form of authority and power. So one would expect them to struggle sometimes, to mutually cooperate at others.

Where did you get that the Abbasids didn't rule by Sharia? Of course they did.