This.

And also, it should still be 12 months, just 4 months (December, January, June and July) should have 5 weeks while the other months have 4 weeks.

The last weeks of December and June and the first weeks of January and July then form a special kind of "half months", where you get a half month salary and pay a half month rent, etc. Christmas and New Years nicely fit in the Jan/Dec two week holiday, which would be 15 days instead of 14.

In a leap year, the June/July half month would be 15 days instead of 14

This way each season/quarter is still equal to 3 months with 13 weeks. And a half year is 6 months with 26 weeks.

Someone mentioned seasonal regression, this solution should solve that.

And the irony of this all... This is very close to how our current calendar started before Caesars fucked it up.