I know its been awhile since this question was asked, but why not use a Jewish calendar in conjunction with an astronomy program to check the moon's phase? The day of the Jewish month is simply the age of the moon, give or take a day. So the 10th of the month is waxing gibbous moon. So whatever date someone spits at you, the moon better be in that phase. Also, the Day of Atonement could not fall on a Friday or Sunday so as not to compete with the weekly Sabbath, but it could fall on a Saturday with the weekly Sabbath. Therefore some minor tweeks had to be used in the calendar.
Using this method, you could probably tell within a day or two. My results (if you have to know) was Saturday October 10, 26AD. That would be using the Julian calendar since the Gregorian was not in existence then. However, if it was, I think the difference would only be a day or two in the date. The ten day error was the compound of errors from Christ's time until then in the 1600's. In other words, the Gregorian calendar adjusted to keep in step with the seasons back then which agreed. That's why the "orthodox Christmas" falls two weeks later. They still use the Julian calendar.