We could go partial French Revolutionary, and have 36 10-day weeks at +3-1+3-3, plus an intercalary partial week at -5 or -6. Since we're not [all] French, the days of the week would be renamed as follows: Workday, Bluesday, Grindsday, Humpday, Schmerkday, Slogsday, Highday, Lazyday, Gamesday, and Chillsday. This would be referred to as the "Bactrian week", and the previous +5-2 week a "Dromedary week". The partial week shall be "Festivus Week".

For essential personnel, they cover all days of the week by working a rotating schedule of 9-day cycles (+3-1+3-2) plus one vacation week of 9 days off in a row, which adds to the previous 2-day weekend for a total of 11 consecutive days off. Festivus has staggered 3 on, 2 off, at double pay. Employees draw straws at 60% probability for working a leap day.

There's no need to rename the months. Years start on January 1st and follow Gregorian leap day rules.

https://en.wikipedia.org/wiki/French_Republican_Calendar

~

I don't know why, but the idea of just dropping a day, though plausible, cracked me up.