Dear David,
This is just a reflection on your question:
Since you assume that the curve is parametrized by arc-length, applying Plancharel's formula to $f'$ yields $$ \sum n^2|a_n|^2 = 1. $$ Moreover, you also assume that the map $f' : S^1 \rightarrow S^1$ has degree 1 and Brezis's formula for the degree of a $C^1$ map from the circle to the circle (Google Kahane's paper Winding number and Fourier series for the formula and the amusing story behind it) yields $$ \sum n^3|a_n|^2 = 1. $$
Averaging these two equations and assuming $a_1 = 0$ one gets that $$ \sum_{|n|> 1} {n^2(1 + n) \over 2}|a_n|^2 = 1 \ {\rm and } \ \sum_{|n|> 1} n^2|a_n|^2 = 1 , $$ but I don't see right now if this and could lead to a contradiction with $\sum n^2|a_n|^2 = 1$.
I don't know if this helps with your precise question, but I think Brezis's formula for the degree in terms of Fourier coefficients could come in useful.
