> calling their looted artefacts by their dynasty (i.e “Babylonian” or “Sumerian”)
There is no "Sumerian" dynasty nor even any similar concept. There is a language we refer to as Sumerian. We don't really know anything about Sumer; we took the name from the traditional royal title "King of Sumer and Akkad".
I don't see what your problem is with identifying artifacts by the period they come from. First, that is significantly more informative than identifying the modern region occupying the same area. Second, no one is going to be confused over which modern country holds the site of Babylon. Third, as you might have noticed me alluding to, "Iraq" did not actually exist at the time.
Who do you think should have the strongest moral claim to royal artifacts from the 23rd dynasty of Egypt? Does putting "Egypt" in the name help answer the question?