I've been thinking about it, and we really don't need to do a realisti quincuncial projection. Instead of making a square project to a hemisphere, I could project the circle inscribed in that square, which wouldn't have any distortion bias. This will mean, though, that at the corners of the square map, the visual planet horizon will be off as much as 26 degrees.

By my math, orbiting the planet along the diagonal will take 41% longer than the horizontal. I don't know if this will be noticeable or not. If it is, we have three options:

1) Ignore it and let the orbit period be inconsistent.

2) Define the equator to be something between a perfect circle and perfect square. This will reduce difference in length in proportion to the amount of distortion it introduces.

3) Artificially boost the speed of ships orbiting along the diagonal vector. Minecarts go 41% faster along the diagonal, too, and people don't seem to think much of it.

If we do anything, I think 3 is the best course of action. Or, we can do a combination of 2 and 3. 20% stretch and 20% speed boost.

The sun complicates things as well, but if we make the planetary rotation happen along either the diagonal or the perpendicular, it should work out.