According to chabad.org,
Beitar fell on the 9th of Av of the year 3893 (133 CE)
the dead were buried 15 years later on the 15th Av of the year 3908 (148 CE).
See here for further background and some of the Talmudic sources here.
Nitei Gavriel Aveilus1 75:6 says that the grave should be dug by a Jew. In the footnote he says that a Jew should be the one who operates the tractor that digs up the ground.
There is no mention of any disagreement on that.