The arc length is now more difficult to compute, because the dx/dt derivative has an extra (v-r) term. You can plug the integral into Mathematica and it can probably do it for you (I could probably do it by hand by collecting like terms but I'm lazy as hell):
It's the integral from t = [0,2*pi] of sqrt((r*sin(t))^2 + (r*(1-cos(t))+(v-r))^2).
Right, and that's called "complete elliptic integral of the second kind", it is not expressible in elementary functions, but, to the first order of approximation, you should get something like v*(1+x^2/4), where x is the ratio of pedal speed to bike speed (~0.185).
Tried to do the series based on formulas from wolfram and wikipedia, I keep getting 1+x^2, which seems to be too large and incompatible with my upper bound from post #10. Either I'm making a mistake somewhere or one of the formulas is wrong.
Edit: found my mistake. To get the first nonzero term in x correctly, I had to take three terms from the expansion of E(k), not two. The answer is, in fact, 1+x^2/4.