How to derive cubic cost function from a problem of constrained optimization?
Given a Cobb-Douglas production function your cost function will be cubic exactly if $\alpha + \beta = \frac{1}{3}$, and in this case you will have $a = b = c = 0$. There is no going around this. If you want to have the kind of cost function you describe you will have to slightly alter your Cobb-Douglas function, e.g.
$$
f(k,l) = (k-1)^{\alpha} \cdot (l-2)^{\frac{1}{3} - \alpha}.
$$