If your objective is to make the simplest possible perpetual (mechanical) calendar, there are several things you can do besides constructing a complicated mechanical computer.
Redefine The Calendar so that all months are the same length. The French Republican Calendar for instance gives each month 30 days. This requires no mechanical complexity at all! It turns out I may have this kind of calendar, in the form of a beat up French verge movement dating from about 1795. The calendar on the movement is simple, (the complicated stuff is quarter repeating work). I’m curious why it doesn’t show decimal time…
Use one gear with lots of teeth, instead of lots of gears with few teeth. A single gear with 1,459 teeth could be indexed once a day, and always show the correct day and month. (1,461 = 365 + 365 + 365 + 366) That’s a lot of teeth for one gear, forcing the gear to be huge, or the teeth to be very small. One way out of this is to make the gear flexible, for instance to make it out of a long closed paper ribbon. The paper would be advanced by perforations (like old film), and it could be coiled up so as to take up less space.
Making A Perpetual Calendar Watch 