Can I restate the problem to see if I understand it?
Your challenge is this - build a time keeping device where the 12 Hours (or 720 minutes or 8640 seconds) at night are longer or shorter than the 12 hours (etc) in the daytime. The time variation is determined by the seasons. Right?
If you have a fixed location or a country with smallish latitudinal (vertical) dimension like Japan (roughly 30° to 45°) I can see how you could calculate (or estimate) average changes, but if the country is like the US, with a range from 18° (Hawaii) to 71° (Alaska), the day/night length changes are unreasonable to calculate. You have too much variation, I think.
Of course if you are on the equator, seasonal change is minimal. Only a few minutes day/night variation across a year.
Maybe that fancy "Key of Time" Hublot watch with the faster-slower buttons could provide us some clues?
PS - Variable speed mechanisms might resolve the problem experienced on a ship, when you travel through time zones - say New York to London or vice versa. Since the day is effectively slower or faster by about one time zone per 24 hours of sailing, and the direction of time variation is dependent on sailing speed and direction, this is just a faster change than the seasons, but conceptually it's a similar challenge.