The earliest Jewish source for the pigeon cure is in the Divrei Yitzchak in 1896 where he mentions that people do it.
The oldest actual source is the classic occult/magic textbook by Frances Barrett called The Magus, or Celestial Intelligencer, published in 1801. He describes the same “cure” using a duck.
(That totally makes this Darchei Emori, by the way. You’re welcome, DY.)