Build rotational Hamiltonian based on Lagrangian of general form

I've been told that one could build rotational Hamiltonian based on Lagrangian of general form: $\mathcal{L} = \mathcal{L} (\vec{\Omega})$. By introducing Euler angles one could rewrite Lagrangian in terms of Euler angles and their derivatives: $\mathcal{L} = \mathcal{L} (\vec{e}, \dot{\vec{e}})$. Angular velocity is expressed in terms of Euler angles in a following way: $$\vec{\Omega} = \begin{bmatrix} \sin(\theta) \sin(\psi) & \cos(\psi) & 0 \\ \sin(\theta) \cos(\psi) & - \sin(\psi) & 0 \\ \cos(\theta) & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix}.$$ (Vector $\begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix}$ is denoted by $\vec{e}$. ) Big problem that I encountered is that angular velocity depends on general velocity vector ($\dot{\vec{e}}$) in a linear way (by linear operator $\mathbf{M}(\vec{e})$ matrix of which is presented above), so general momentum $\vec{p_e} = \frac{\partial \mathcal{L}}{\partial \dot{\vec{e}}}$ is not an explicit function of general velocity vector($\dot{\vec{e}}$). That is Donkin's theorem or Legendre transform cannot be applied because general velocity components cannot be explicitly written as functions of $\vec{e}$ and $\vec{p_e}$.