Derive the cost function for a Homothetic production function
There must be an error in your proof.  Where does your proof come from?
The part "for the above to true it is clear that the ratio must be independent of $w$," cannot be True.  Your ratio must be independent of $q$ but not of $w$.
Take for instance the Cobb-Douglas case (which corresponds to a homothetic production function) for a counter example. See Diewert or Chambers for a proof of your result:
Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press.
Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2.
In the Cobb Douglas case, the production function $$q=x_1^\alpha x_2^\beta$$ yields the first order conditions $$\frac{w_1}{\lambda}=\alpha\frac{q}{x_1}$$
$$\frac{w_2}{\lambda}=\beta\frac{q}{x_2},$$ so that optimal input demand satisfy (in your notations) $$\frac{x_1}{x_2}=\frac{\alpha}{\beta}\frac{w_2}{w_1} \equiv \frac{c_1(w)}{c_2(w)},$$ which is independent of $q$ and implies (show it) that $$x_1=H^1(w,q)=c_1(w)b(q),$$ $$x_2=H^2(w,q)=c_2(w)b(q).$$ It follows that the cost function $$c(w,q)=w_1H^1(w,q)+w_2H^2(w,q)=a(w)b(q),$$ with $a(w)=...$. A similar derivation holds for the more general homothetic case, because by definition of homotheticity, the production technology can then be written as $$y=f(x)=g(h(x))$$ where h is homogeneous of degree one...