Does the minimum power required for level flight depend on CD0?
Thank you for the excellent question!

The whole expression is only valid for the optimum polar point where the ratio $\frac{c^2_D}{c^3_L}$ reaches its maximum. If you deflect spoilers, this optimum will shift to a higher lift coefficient, so both aircraft will not fly at the same lift coefficient or the same speed any more! Always keep in mind that in this special condition the lift coefficient can also be expressed as 
$$c_L=\sqrt{3\cdot\pi\cdot AR\cdot e\cdot c_{D0}}$$
Just plug that back into the equation and the minimum power expression will make a whole lot more sense:
$$P_{min}=\frac{4}{3\cdot\pi\cdot AR\cdot e}\cdot\sqrt{\frac{2\cdot W^3\cdot\sqrt{3\cdot\pi\cdot AR\cdot e\cdot c_{D0}}}{\rho\cdot S}}$$
This is still the same equation, but now it should be obvious that opening the spoilers will require more power.