Not a joke. With a decent fission power system and a plasma drive, speeds of up to 0.1% the speed of light (speed of light, C, is 300,000 km/sec, so 0.1% of that equals 300 km/sec) should be feasible. (Probably 150 km/sec is probably more conservative and reasonable, but we don't know how well we can engineer it until we try.) At that rate, since the system is only 20.5 light-years away the probe will arrive in only about 20,500 (20.5 * 1000) years.

I imagine that the probe would leave Earth orbit with a mass of some 10 tons, of which perhaps 2 tons would be the reactor and thermoelectric power converter. Another 4 tons would be rocket and propellant. This would leave the probe with 4 tons for payload.

No, I don't expect it to be working when it gets there. However, it would be a start. The probe could do some very important astronomy while it lasted by giving us a much, much larger baseline for parallax measurements. Even if it contained a telescope with a 1 meter mirror designed solely to determine stellar positions, the baseline would be so much larger, perhaps 1000 times as large as the current one, which is a mere 2 AU, or roughly 300,000,000 km.
This would greatly extend the good work performed by the Hipparcos satellite.

At 0.1% C, that means that it would take the craft only one million seconds to travel 2 AU. One million seconds = 1,000,000 / 86400 days = about 12 days. If the probe survived 6000 days (~~ 16 years), this would mean that the probe would have available a parallax baseline distance 500 times that available at present. Simple trigonometry shows that we would have absolute distance measurements of well over 200 times as far as currently possible. This would establish a new meterstick for the Universe far more accurate than the one we are now using. Given the record, there is good reason to believe that a probe could survive for 16 years.

Communication would be by laser rather than radio, since the range and bandwidth of a laser signal would far exceed that of a radio signal with the same diameter antenna. Communication would be one-way. The probe would be programmed to perform a full-sky survey each and every year with its gradually larger baseline.

Crazy? Perhaps. But worth considering.