Your logic makes a number of errors.
1. Scientific exchanges have nothing to do with this topic. Gunpowder is not relevant.
2. The republic does not concern Iran. Iran is a "Jomhuriye Eslami" taken from Islamic technical texts, roughly translated into Islamic Republic to make foreigners, or simply dumb people understand what the political structure could encompass.
3. Islam was not influenced by 'persians' or 'arabs' since there was no such uniform identity as we know today. Both of these civilizations were barely any more advanced than the other. Iran was a feudal society with a minority of aristocratic Sassanian rulers who prefered warfare over scientific achievements. Just because Arabs were even worse off, killing their daughters does not make one better than the other. To make the suggestion that Islam was "heavily" influenced by pre-islamic innovations is to deny their regressive postures.
In Islam, cultures can be sustained, only if they are acceptable by Islam. In pre-Islamic Arabia, many things were done in their culture deemed Haram, and thus had to be abandoned, no matter how dear that culture was for them. As Rumi himself said: Speak Persian, but Arabic is sweeter. Like today, "Persians" and "Arabs" did not have a sense of being two seperate entities, divided and at war with each other, these words and their meanings did not even exist in the way we know them today. These peoples of different tribes who all migrated to the middle east appreciated each others culture, and borrowed extensively, but consistently after Islam since this was the 'breathing space' for what was to be a 400 year golden age.
Aside from that, Shari'a does not depend on Quran alone. There is no such thing as an exclusive "Quranic law" nor is this practical.