Are we not space cadets?

First step would be gathering requirements, particularly the required precision and accuracy. The "keyhole" is described several hundred meters wide, suggesting precision a couple of orders of magnitude smaller: Say a meter or so, across tens of millions of klicks. What's a few parts-per-billion? The implication is that ranging will require shorter wavelengths, e.g. visible light lasers.

That means a reflector, though maybe not spent CDROMs....

I went and took a shower, and as often happens, that damn telepath somewhere beamed a notion into my head: Lasso the thing and garland it with mylar reflectors.

Imagine a long tether. At its head is an impact anchor a la DeepSpace1, augmented with explosives that go off just after impact, mushrooming the impactor to produce a molly bolt effect and hopefully a secure grip. Trailing behind on the tether is a series of smaller impact anchors.

I'm assuming that the asteroid is tumbling. Assuming good guidance and nav, the lead impactor hits dead center. It transmits a pulse back up the tether to the "caboose", which fires a short burst of gas to deflect the tail away from dead-center. After the lead impact, the tether will swing around past the asteroid, pivoting on the head anchor, and whip back around to wrap itself around the asteroid. As each of the smaller anchors impacts, it fires a bolt into the surface to anchor the cable at several places. Each small impactor could have a short counterweight, so that centripetal force automatically points the impactor at the asteroid surface.

Once the tether's wrapped around the rock, it can begin to unfurl a series of mylar refecting panels (aha--furled, they're the counterweights for the anchors). Again assuming the rock is tumbling, centripetal force will keep the panels fairly rigid. I'm not sure if they should be sheets, or a shape with a triangular cross-section to ensure that one surface is always available for a reflector. The odds improve the more panels are unfurled, resulting in something that resembles a windmill that crashed on an asteroid. If the tumble is simple, the lasso will be wound around the virtual "equator". If the tumble is complex, that's actually better, because then the tether will wrap more of the surface, more like a ball of twine, and deploy reflectors at several "odd" angles.

OK, what have I gotten wrong?

That damn telepath, here's another: A package like a lightsail, perhaps disc-shaped and spun up, with counterweights along the perimeter. When it impacts, it should fold around the asteroid like wadding aluminum foil around a potato. How to anchor it?...hmmm...velcro? Maybe somebody with small kids can talk about the state-of-the-art in nappy fasteners...similar topological challenge, isn't it? Same need to wrap a moving target with split-second accuracy, too...;-)