Interesting idea, but no - my model is Alpha Centauri and both Centauri I and II output most of their light in the visible spectrum. Proxima Centauri (Centauri III) is mostly infrared, but to a planet in orbit around either of the main stars it would appear only as a mere magnitude 2 star and dimmer than Sirius or even our sun despite both being light years away.
An eclipse requires the occulting object to have a visible disk, which neither asteroid has, and that that disk have an apparent size equal to or greater than the blocked star. Carthasana's primary moon, Balcra, has this property like our own moon - it is exactly 40 times smaller than the sun, but 40 times closer. The cold sun can also be eclipsed as it just barely has a discernable disk to the unaided eye about 1/40th to 1/20th the size of the sun.We also had another thread talking about eclipses too. With a binary and two asteroids you must have them frequently. Perhaps with an 80yr orbit those eclipses could last longer than ours and a 10hr asteroid might have quite sudden light dropping eclipse effects lasting mere seconds...
No. BTW, my reference on both the asteroids is the fact I've observed both with an unaided eye and with a telescope our own International Space Station. The magnitude given is brightest possible, a good deal of the time the station is much dimmer. The asteroids are a bit further off though and unlike the ISS they orbit out far enough to escape being eclipsed themselves at least on some nights, though they do wink out at night as they get eclipsed. However, when the cold sun is out when this happens they just fade away, lost in the scattered blue light created by the cold sun.If the seconds star at mag -20 is out by itself then presumably you would be able to see that asteroid as a dark shape going across the sky.