Generally no. The Euler-Poincare equation derives all of its structure from the Lie bracket, and an arbitrary constraint does not respect this structure. The most simple counter-example is probably a subspace constraint. Let $V \subset \mathfrak{g}$ be a subspace which is not a sub-algebra. This subspace generates a constraint distribution on $TG$ in the obvious way, which we'll denote by $VG$. The constrained Euler-Lagrange equations on $TG$ are given by

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{g}} \right) - \frac{\partial L}{\partial g} \in (VG)^\perp \\ \dot{g} \in VG$$

Here $(VG)^\perp \subset T^*G$ is the annihilator of $VG$. These equations can be reduced to $\mathfrak{g}$. Here they take the form

$$\frac{d}{dt} \left( \frac{\partial \ell}{\partial \xi} \right) \pm {\rm ad}^*_\xi \left( \frac{\partial \ell}{\partial \xi} \right) \in V^\perp \\ \xi \in V$$

Here $V^\perp \subset \mathfrak{g}^*$ is the annihilator of $V$. In anycase, the above equation is not an Euler-Poincare equation, and It can not be transformed into one (although, one may call it a "non-holonomic Euler-Poincar'e equation).

Generally no. The Euler-Poincaré equation derives all of its structure from the Lie bracket, and an arbitrary constraint does not respect this structure. The most simple counter-example is probably a subspace constraint. Let $V \subset \mathfrak{g}$ be a subspace which is not a sub-algebra. This subspace generates a constraint distribution on $TG$ in the obvious way, which we'll denote by $VG$. The constrained Euler-Lagrange equations on $TG$ are given by

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{g}} \right) - \frac{\partial L}{\partial g} \in (VG)^\perp \\ \dot{g} \in VG$$

Here $(VG)^\perp \subset T^*G$ is the annihilator of $VG$. These equations can be reduced to $\mathfrak{g}$. Here they take the form

$$\frac{d}{dt} \left( \frac{\partial \ell}{\partial \xi} \right) \pm {\rm ad}^*_\xi \left( \frac{\partial \ell}{\partial \xi} \right) \in V^\perp \\ \xi \in V$$

Here $V^\perp \subset\mathfrak{g}^*$ is the annihilator of $V$. In any case, the above equation is not an Euler-Poincaré equation, and It can not be transformed into one (although, one may call it a "non-holonomic Euler-Poincar'e equation).