Ooops. I'm afraid I can't help, here. :-|
Great ! Clever test. I agree, you can just use vprime there.
That the difference is small may be due to the fact that in principle vr_mean is supposed to be small. In contrast, r*Omega should be way larger than vt_prime, so I expect removing the + r * omega in the calculation of vt_prime will have more effect.