Notwithstanding this, your reservation about the existence of a binary stellar companion requires exploration.
our next nearest star is Sirius
Sirius is 8.58 light years from Earth. Alpha Centauri C is 4.24 light years from Earth. However, this star is one of a three star system which could cause eccentricity of orbit, particularly at periastron. A doubtful candidate.
With no observable star as a logical companion, perhaps it has ceased to emit light. Or is travelling towards Sun at a speed which fits the parameters.
the mystery DarkStar object to be within our SolarSystem - closest possible to Earth..
An intriguing suggestion.