Maybe it’s on a highly elliptical orbit within the star itself having been captured at some point? That way it would spend only part of the time near the core and the rest farther out where it’s effect is less.
If that were the case, though, the star’s fate is a matter of if the black hole will dissipate faster than it is absorbing matter. If it dissipates, being too small, the star benefits from the relationship because of what stirring of its mass has occurred. If not then the star is ultimately doomed by the parasite now within it.
The test of such a theory might then be if, over time, the dimming effect increased (star doomed) or diminished (star benefits) as neither would seem a likely outcome of an orbiting cloud of stuff.
There is some indication that Tabby’s has been dimming at the rate of 0.16 magnitude per century (http://www.skyandtelescope.com/astronomy-news/tabbys-star-weird-star-gets-weirder/).
The discovery of another star similar to Tabby’s so soon after Tabby’s was found may indicate they’ve found another class of stars, in this case, stars with small black holes in them, stirring up the pot and opening a new window on stellar cores that could keep astronomers and physicists busy for some time.
I love the possibilities: primordials on long elliptical orbits, dissipating while outside the star, perhaps fully replenishing themselves when they dive back into it, binary primordials whose link to one another is broken by the star, all happening in every class of star at every stage of evolution, etc.
(Love your tagline!)