The Calormen were not Muslim. Arab, yes, and there is an argument for racial insensitivity in the books, but they believed in made-up gods Lewis invented - not a counterpart to Allah. Although they played the part of villains in many of the books, they were often shown to be misguided over evil. C.S. Lewis was a famous apologist who took his faith in Christianity very seriously, and although he did say there may be truth in other religions, why would he claim any other religion to have equal legitimacy to Christianity?
As for your fictional religion, I do see some parallels with some religious values and subsequent hypocrisies but there's nothing actually religious in this society as you describe.
Where do they believe they got their way of life from? Are they polytheistic? Monotheistic? Other? What rites and rituals are associated with this religion? What is their traditional background? Who watches over the religion? How old is the religion? How integrated is the religion with history?
Questions you might want to think about. Or, to prevent world-builders-disease, answer the three you find most important and write a few scenes for your story where religion plays a role.