Banking and Rationalism
Some time between 300 and 1200, Europeans advanced from wild tribes, who in most battles would easily be decimated and who were usually unable to achieve success if they hadn't the element of surprise on their side, into a real, though barbaric, civilisation, with pretty disciplined, well trained and very well equipped armies. I suppose there were several reasons for that - general development all over the world, increased population numbers from more agriculture and maybe climate change, the one or other modernising efforts, the high frequency of infighting in Europe (competitiveness between the countries), and so on.
But afaik, the relative decline of the Muslims and rise of the Europeans really started in the 15th/16th century, when, out of the blue, city states like Venice (with backing by other European powers) were able to muster fleets superior to everything the Muslims had to offer. Why? Imo, the Renaissance brought back the same kind of rationalism which had made Greece a world power for some time, and the invention of more advanced banking systems made it possible to concentrate the efforts of the people more than ever before - for instance into creating large fleets. The Muslim world had seemingly stronger powers for a few more centuries, but it was no serious competition for Europe anymore from around 1800.
A POD would probably be possible at any time - though an early one would obviously make it more likely to result in something really big.
I cannot imagine that there were no attempts to create banks, to get more rationality into the populace, and so on. But somehow, they seem to have failed mostly after a golden age ending in the 16th century. I don't know Muslim religion enough to decide what exactly went wrong, especially as literacy and knowledge are actually in high regard among Muslims. I suppose one reason is that more and more knowledge spread which was incompatible with their religious believes. Catholics mastered this problem by separating state and church, while Protestants solved it by seeing riches gained as divine vindication of the ways of an individual and being quite open about how God runs the universe. The fact that Mohammed was a warrior, probably also helped a bad development, as men who were not soldiers or leaders, or who acted similarily tough, supposedly had difficulties being accepted in their societies (I can't tell, though, due to lack of knowledge - the exact social mechanisms might be different or more complex). Also, the territories in Muslim occupation were usually pretty dry and, due to climate change in north Africa and the Middle East, getting even more dry at that time. A hot climate is not really good for industriousness, and less fertile grounds means less population growth.
The water problems could have been solved with more intense irrigation, more use of salt water (creating artificial fishing lakes, for instance), and similar measures - but it would demand a lot of rationality for them to be done in the right magnitude. A Muslim might be able to tell whether such measures would have given the according leaders and workers fame and respect.
The heat problems could have been solved with the right architecture - though that only helps with factories, not with quarries and the likes. It would also require that leaders care about their worker's condition and productivity - which in militaristic societies is not necessarily a given.
No interest rates in banking and less options to get around such limits were also an important factor. A simple way around this would probably have been a stock market for businesses and a mix of rent and payback (leasing) for consumers (especially house building) and small businesses. A lot of rationality and legal protection would be needed to create and maintain such a system, though. The riches some people would aquire through this would also have to be protected from both politicians and fighters, while the interests of the "small people" against such businesses would also have to be protected. I'm not sure Muslim religion at that time would have been very good at that. From evidence, it seems not so, despite some steps in the right direction.
What also astonishes is that apparently, knowledge was not kept as sacred as before the 16th century from that time on. I'd usually expect many more libraries like the ones in Granada and Cordoba to continue being created in the Muslim world - actually being used for learning and reference, extended with a larger number of new books every year. Not to mention universities, scientific laboratories, and so on. I suspect people of knowledge who had opinions different from the religious leaders were considered a threat by the latter, too.
There also seems to have been a reluctance to copy any European advantages - like having a blacksmith in every village from late medieval times, to the progress coming during the Renaissance. It seems to me, the Muslims considered themselves too superior to Europe to even register such things.
Another European advantage probably was an upcoming (mostly protestant) ideology of avoiding luxury and waste. After the plague, the survivors had a very high standard of living. They could have used that for luxury, palaces, food, and so on. Instead, many saved their riches, which helped banks and governments, and allowed concentrating efforts not only through space, but also through time, by having the results of the labour of many years available in times of need. Muslims seem to have spend quite a lot on palaces, art work, gimmicks, and so on, instead.
A simple POD to get around all this might be for some Muslim scholar to publish a theory that due to general advances, the preparations necessary to win a war become more and more complex. It might show in detail what kind of spending helps get stronger (irrigation, arms production, learning, trade, saving barter to get help/arms when needed...) and what just wastes time and effort (unnecessarily elaborate building, luxury...). This would both create a sense of progress and help concentrate efforts on expansion. The theory should also have a paragraph detailing that rules must be followed with flexibility - an animal skin might be a luxury in the desert, but essential winter wear in the mountains, for instance. That would help expanding into different climate zones and avoiding blind dogmatism. It might even help reform core Muslim believes over time. Ideally, it would also include a chapter of how to use information about past battle preparations to find what made the winner win, besides religion, morals, and so on, and some general information on when a fast and impatient approach is likely to succeed, and when systematic and cool headed measures are better. Both with pragmatism, of course. Under the shock of the reconquista, a Muslim leader propagates such a book.
The result might be a Muslim society avoiding showing off riches, avoiding wasting riches on short term benefits, and so on. A Muslim society concentrating not only what a leader can gather at a certain moment, but the potential of the whole society he rules. A Muslim society not only learning to put the productive efforts of several years into one battle, but also, after some time, to spend more effort into getting stronger internally than into expanding.