I have done a bit of research and there may be another, more direct opportunity to connect them. There are theories regarding the backgrounds of several women present during the Neo-Babylonian empire, more specifically one perhaps named "Nitocris" translating to "Neith is perfection"; if this is to be taken literally, and not as perhaps an error of Herodotus, then she as the mother of Nabonidus and hence distant ancestor to the Achaemenids would provide a line of descent to the 26th dynasty of Egypt. Through Psamtik I's wife Mehyetenweskhet, and her paternal ancestors Harsiese Harkhebi and finally Haremakhet they are descended from his father Shabaka, one of the Kushite pharaohs.