Regarding a Solar System occulter:
The occulter would also be in orbit around the Sun, so it would not occult the star once every Kepler orbit. Anything in our Solar System occulting the star would likely occult other nearby stars not because it’s big but because it’s moving. There’s no reason it would come back exactly to this star and only this star, from Kepler’s perspective.
There are events occurring throughout the 4 year Kepler mission, not just a few events spaced by (roughly) two years.
The “exactly 2 years” bit is a red herring — Kepler does not orbit the sun every year or every two years, so there is no coincidence to explain.
Anything with a sharp edge in the solar system would create diffraction when occulting the star, which we do not see
The timescales are all wrong: it takes days for the star to get dimmer, which suggests a very low mutual velocity for the occulter and the star, implying it is in orbit around the star.
The “soft” ingress and egresses of the star means you have to invoke a “fuzzy” absorber without sharp edges. It would presumably be visible as a dark part of the sky in deep imagery of this part of the sky.