“Are there any frum doctors in Brooklyn or Queens doing tests on a motzei shabbos?”
The immigration authorities do NOT require a test from a FRUM doctor or nurse. If you can find a qualified galach administering the tests, thats fine. They don’t care about the hashkafah of the testing firm. There are also 24 hour tests being offered now at JFK and the urgent care near the Belt Parkway by the JFK entrance.