Yes, there really was a FAL in 5.56×45. It was the SAR-4800 made by Imbel in Brazil and imported by Springfield Armory. According to Ian McCollum of Forgotten Weapons, only about 700 were imported into the United States and these came with those ugly post-ban thumbhole stocks.
He discusses the history of it while examining one that was converted to the FAL Para configuration. While it probably is a bit heavy, it is still pretty cool.