Having a “remainder day” is weird, but it’s hard to avoid. It really sucks that 365 doesn’t divide nicely into much at all. 5 and 73 are the only non-trivial answers. five 73 day months? Can’t even call it a month at that point.
I guess 13x28 + 1 does indeed make most sense…
speaking of leap days, I also considered using a quad-year as a unit, integrating the leap day as a standard day. 365.25x4=1461. But that only divides by 3 and 481, even worse!