Is the sleight of hand you are referring to in regard to the fact that the 3rd Babylonian month (Simanu) matches up with May/June of the Julian Calendar.....not July? Therefore a lunar eclipse that occurred on July 15, 588BCE would be irrelevant?
I'm no calender expert but the eclipse occurred on the 15th day of the Babylonian month. That most likely matches to July 4th (correctly I presume?). Then they say an eclipse occurred on the 15th day of the Julian month in the year they want (588). Most will carelessly read: "oh, the eclipse was supposed to occur on the 15th... in 568 it occurred on the 4th but in 588 it occurred on the 15th. Clearly 588 is correct"
If that speculation is true that's really a dishonest way to write that paragraph.