As the article linked to by Angielczyk states, the earliest recorded use of the Star of David as an unambiguous peculiarly Jewish symbol is in Europe, circa AD 500. If it was used earlier as a religious symbol, there is no record or mention of it that has survived. It is recorded in earlier Jewish contexts, but it always seems to occur as a decorative pattern rather than as a symbol (eg. the repeating pattern of hexagrams, pentagrams and swastikas in the Capernaum synagogue). It does not appear to have been a "Jewish symbol" during the time of the Exile.
Finally, the ruins of Babylon have been worked, reworked, shoddily rebuilt by Saddam Hussein, blown up by ISIS and otherwise quite thoroughly picked over by archaeologists and looters for the past several hundred years. This would have been true in 1980 as well, except for the ISIS part. I'm pretty sure there are not going to be any genuine artifacts of any kind just lying around for people to pick up, let alone artifacts of such cultural and historical significance as physical evidence of the Exile. I could be wrong, but I believe the only evidence of the Exile found in Babylon itself have been a few clay tablets mentioning Jewish families.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis