The Mexico story is true. It was during Prohibition and the distillers were Joe Beam and his son, Harry. Many of us know Harry's granddaughter. The old pot still they used down there is now in the possession of Vendome, in Louisville.
Obviously, the brand came back after Prohibition, but not at the original location in Anderson County. The Dowling family, which had owned the distillery and brand name since 1903, built a new distillery in Anchorage, a suburb of Louisville, after Repeal but sold it soon after to Joe Makler, a Chicago whiskey broker, who moved the name to a distillery he owned in Bardstown.
That last distillery to be called Waterfill and Frazier had been built in 1850 by the Lancaster brothers. It became part of the Whiskey Trust in 1903. It came back after Prohibition as an independent, called Independent. It briefly bore the Shawhan name, but none of the Shawhan family were involved at that point. The owner was Tom Pendergast, the political boss of Kansas City, who also owned the distillery today known as McCormick. After WWII, it was sold to Makler, who operated there until 1969. Jim Beam bought it in 1974 but only wanted the warehouses. The rest of the buildings were demolished but the warehouses are still there and still used by Jim Beam.
At some point, Heaven Hill acquired the Waterfill and Frazier name, but I don't know if they're using it. They also own the Dowling name. It may have had an interim stop between Makler and Heaven Hill, but your bottle is probably a Heaven Hill product.