And keep the named 7-day week cycle as-is independently of that, this one has been going for millenia and we need some weekends after all
Actually, looking it up now it looks like the usage of the name "Julian" is due to the way the old tracking systems were programmed and used "JDATE" or similar. Nowadays it's just burned in habit that it's called the Julian Date.
My guess is that the idea of counting days attached the "Julian" name to day number in a year.
year/day/hectosecond.second(.millisecond and so on to the desired precision)
The current date-time, to the second, would be 2023/308/342.65
The days, like the years and second-based units, are 0-indexed.
There are 100 seconds in a hectosecond, and normally 864 hectoseconds in a day.
And to go with it, I'd advocate either a "3-on, 2-off", or "6-on, 4-off" work cycle, depending on the nature of the work, or even just personal preference. This way, today(308) would obviously be on a "weekend" either way.