I'm not sure if this kind of distinction would be useful to historians. "European" is a geographic designator with lots of political significance, but tells us nothing about culture or society.

The world does not change unrecognizably at the Bosphorus: in the famously cosmopolitan Ottoman Empire there were substantially Greek, Armenian, and even Slavic-speaking communities across Anatolia and other "Asian" parts of the Empire. These communities were generally Christian as well. In European provinces both Muslim and Turkish communities were very commonplace, persisting to the present (with grave international significance) in Bulgaria, Bosnia, and Albania.

I find it interesting that you chose the word "dynasty", not empire. As a dynasty the House of Osman certainly did not originate on the European continent but for centuries they ruled a great portion of it. Searching for general catalogues of European royal families, I did manage to find Lars Ulwencreutz's 2013 book The Royal Families of Europe. The writer, who appears to be an amateur historian from Sweden, reserves a chapter for the Ottoman Empire. In doing so, Ulwencreutz suggests that the Ottomans should be considered a European dynasty. Some older documents, such as the 1904 American Almanac, Year-Book, Cyclopedia, and Atlas also include the rulers of Turkey in their lists of European rulers.. However, this was not a unanimous opinion. While this concerns the government at large more than it does the ruling family, international relations theorist Yannis A. Stivachtis argues in Turkey-European Union Relations: Dilemmas, Opportunities, and Constraints that turn-of-the-century Western observes described the Ottoman state and legal system as separate from Europe, most notably in the 1907 Hague conference wherein the Ottoman Empire was denied a full seat at the arbitration table on account of their supposedly less-than-European traditions.

These are not scholarly historical texts, but whether or not the Ottoman dynasty is European is not the kind of questions historians can answer. The best we can do is assess current opinions of the Europeanness of the Ottomans and understand historic material concerning popular attitudes toward this question.