Given the nature of your project, it would not be unreasonable to widen the range of applied technologies. If you have FPGA's, microcontrollers, and vacuum tubes, then why not relays and gears as well?
Anyway, clever methods for constructing astronomical clocks, orrery mechanisms, and such like have been around for awhile. Classic examples, that include Moon dials, include the 1500's era Astrarium of Giovanni de Dondi:
All the way back to the Antikythera Mechanism, which may owe at least part of it's design to Archimedes. Some beautiful amateur build sites, heavy with technical details, can be found at:
Speaking of things that are reliable... has there been a temporary suspension or reorganization of the designlines newsletters? Mine had suddenly stopped arriving... about one month ago. ALL of them. At the same time.
@Bellhop: It's common for Grandfather Clocks to have a "Moon Dial" that shows the phases of the moon.
Ah Ha! Now I see what you mean -- I tell you, they were amazingly clever in ye olden times -- now you have me wondering -- did you see the comment earlier about the Hebrew lunar calander that was calculated using only integers -- this would easily map onto using clockwork gears/cogs.
It's common for Grandfather Clocks to have a "Moon Dial" that shows the phases of the moon. I remember that the actual dial looks like a sawblade with two stylized "Moons" painted on the face. The dial was held steady by friction and a pawl poke up from the clock mechanism (from a cam?) to push the dial a notch each day.
I agree in that my guess is the same as Elizabeth's. Since you can see the full moon dates are more than 28 days apart from the table, the period has to adjust to the approx 1/12 arc movement around the sun and so the moon has further to travel to get back to where the sun is iluminating the face of the moon that is facing the earth (approx 27.3/12 or 2.2 days). Also, you need to start with a fractional day (not Feb 14, but Feb 14.7 approx). This should explain the 3 days difference you were seeing.
Or you could follow the current model and use a 19-year look-up table.... according to Wikipedia, it does lag the Gergorian by a day every 224 years, but then again, the Gregorian is gaining time on the "sol" part, with the equinox dates coming earlier. Even the recent addition of the 400-year "leap skip" doesn't fix that completely.