How significant are the relativistic effects at 0.2c (the probe speed they are aiming for)?
As a rule of thumb, you can approximate sqrt(1 - v^2/c^2) = 1 - 1/2 (v/c)^2. So if v/c = 2, then the correction is approximately 1 - 1/2 .2^2 = 98% as you said. This formula is easier for back of the envelope calculations and the important point is that it's quadratic. (I always forget the 1/2 :(. )
The GP comment used the wrong correction of 1 - v/c, that is not quadratic so the correction is much bigger.
For speeds that are close to c, you have to use another "Taylor" series to get a good approximation. If v is close to c, then the approximation of the correction is sqrt(2(c-v)/c). So at 99%c you get 4.5%.