I'm not in a position to toss real numbers around at the moment, but here's a thought experiment:
Iridium satellites reflect sunlight toward Earth's surface, producing bright flares, up to magnitude -8. That's about 1 million times brighter than the faintest light source a fully dark-adapted eye can perceive -- a five-magnitude difference in brightness is a factor of 100, and under really dark skies humans with excellent vision can see down to magnitude 7.
I think these reflections are coming off a panel of about 3 x 1.5 meters, so it's comparable in size to your probe.
These satellites orbit about 800km above Earth's surface. How far away could one be, and still produce a visible reflection? Brightness falls off as the inverse square of distance, so the greatest distance at which you could barely see a reflection would be 1,000 times greater, or 800,000km.
Light travels at 300,000km/s. A probe traveling at 0.2c would cover 60,000km/s. Traveling 800,000km would take about 13 seconds.
That means that something the size of an Iridium satellite, reflecting sunlight directly toward an observer during its entire passage, could be visible for up to almost half a minute -- blue-shifted for the first part of that, and red-shifted for the last part. (Unless it's quite far away at closest approach, which would make it dimmer, the transition from "basically heading straight for you" to "basically heading straight away from you" would take place in the literal blink of an eye, a matter of milliseconds.)
Now, this is probably an upper bound -- probes won't have the same shape and reflectivity as an Iridium satellite, and won't be oriented to reflect directly toward an observer during their entire transit. But, somewhat to my surprise, it looks like you might be able to see one approaching and receding.
On the other hand, it's also possible that an object making 0.2c might have a visible interaction with the solar wind. (That requires more numbers than I'm able to run at the moment, but having thought on it a little bit, it seems unlikely.)