To make ##q^0## 0, we just need to boost it so that ##|\boldsymbol{k}^{2}=|\boldsymbol{k}′|## ~.
$$q^0=k−k′=\sqrt{m_e^2+\boldsymbol{k}^2}−\sqrt{m_e^2+\boldsymbol{k}′^2}$$
But this is possible regardless of the approximation $m_e\to0$. What does it mean when it says that ##q^0## becomes 0 by...