Derivation long run cost function of three inputs with Leontief-like characteristics
Your reasoning that $L_1=L_2^{1/3}$ is valid for any $K$. Indeed if this equality does not hold you can lower the cost by reducing the excess input of either skilled or unskilled labor. Thus you can rewrite the cost minimization problem in two dimensions instead of three

\begin{equation}
\min_{K,L_1}{200 K + 5 L_1 + 6 L_1^3}
\end{equation}
subject to
\begin{equation}
K^2 L_1 = q
\end{equation}

From the constraint you can replace $L_1$ by $q/K^2$ in the objective function, which becomes a minimization problem with one variable $K$ and a parameter $q$. Solving it will give you the cost function as the value function of the problem.