I am not sure how they computed the period of the distant planet, but I am nearly certain it was not from angular measurements. If they have the period of the near-in eclipsing companion planet and can measure the associated Doppler shift of its star, they provides constraints on the size of the small orbit and therefore mass of the star. Knowing mass of the star and distance to the planet one can estimate the period of a circular orbit.
27,000 years is the number reported in the article.
Even assuming observation from above or below the orbital plain, an orbit that large would be difficult to discern from a straight line over a short period of observation. So your surmise concerning fitting observed data to theoretical orbit calculations makes sense. Once, that is, you believe you are actually dealing with a planet in orbit.
Small as it appears, how big would that planet have to be to register as an image of (?) pixels from 1200 light years away?