Could an object occupy this position, and how?
Yes, and with difficulty.
Of course an object can occupy that position. The trouble is station keeping. Where it is, the gravitational pull of the two stars exactly balance (the radiation and solar wind pressure do not, but that's secondary).
What if the asteroid is nudged a few centimeters out of position by any one of several phenomena (for one, the Yarkowsky effect)? Consider also that the two stars will have different mass loss ratios, so the barycenter is bound to move - it won't stay in the geometric middle.
When the asteroid is slightly nearer to star A than star B, the pull of A will be felt more strongly, and the asteroid will tend to drift even closer to A. At the same time B's pull, that might avert the fall, will become weaker. So, the asteroid will start falling towards A.
To avoid this, you need some sort of attitude control and station-keeping machinery. The most economic way would be to modulate the asteroid's reflectivity and radiancy, in effect converting it into a sort of statite. The asteroid will reflect the weaker red star's light, and absorb the brighter one's, determining a net thrust towards the brighter star; and this must be enough to balance the brighter star's stronger radiant push. If the asteroid finds itself "falling" towards the blue star, it will start reflecting back its light; if it finds itself falling towards the red star, it will stop reflecting the blue star.
To be able to do this, the surface to mass ratio of the asteroid must be pretty high, which means the asteroid needs to be hollow or made out of some sort of foamy stone, like pumice.
Also, of course, it will require a source of energy, actuators, mechanical and electronic command and control systems. In short, it will need to be more like a space station than a simple asteroid.
On the other hand, the Ancient Race might have done this differently - they might have incorporated a sizeable black hole inside the asteroid, so that the two stars are actually orbiting the black hole. This is also an unstable configuration (the two stars will tend to "fall" on each other, unless the black hole gravitational field is somehow both made asymmetric, and made to rotate at the same rate as the two stars, trapping them inside two gravitational "pits"). It can be partially stabilized... but you need more stars to do that. And it will gall the slimy blue blobs from Mars :-)