This has been solved exactly. The linked article gives details. [Addendum] To directly answer the original question for the limit $m_2 \to 0$: Of course if $m_2=0$, then the force between the two masses is $0$.
In the lab frame, the repulsive force looks weaker. There are two ways to explain this, both equivalent: There is a magnetic force between the charges, since moving charges create magnetic fields and parallel currents attract--just like a wire.