Once you know the year's doomsday, how much work is left for a specific date?

Very little — you jump to the nearest doomsday date in that month and count a small offset, mod 7.

Because every month has a doomsday date within a couple of weeks of any target, you never count far. To find, say, 20 December: December's doomsday date is 12/12, which is the year's doomsday; 20 − 12 = 8, and 8 mod 7 = 1, so 20 December is one day after the doomsday. The heavy lifting is finding the year's doomsday once; each date after that is a tiny addition.