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Something you may not have thought of: the Earth-Luna system has the barycenter inside the Earth's radius. Fairly deep inside, in fact -- about 25% of the way down from the surface.

Your system as described is *less* binary-like than our own. Either the moon is smaller than Luna, or farther away, or both.

Options:

- Change the fictional setting. Figure out the desired effect, and engineer it to produce that result.

- Months aren't important at all to your civilization. Seasons (assuming significant axial tilt or wild orbital eccentricity) might be the preferred unit, and there's no reason to arbitrarily claim four seasons. Perhaps there are only 2, or they really like some other number -- even seems more likely than odd to me, but perhaps they have some cultural specialty for 5.

- The moon is small and close and therefore fast: in another million years it will be a disaster, but right now it's a speck that is sometimes visible in the daylight, and has visible motion if you have any sort of reference tree or mountain.

- The moon is large and far away -- so far away that it's in the L1 or L2 point, and is either *always* shading a chunk of the planet (weather effects will be interesting!) or *always* in the planet's shadow and thus only detectable by occluding other astronomical objects (mythological effects will be interesting!).