How can an attractive (or less repulsive) Lorentz force between two charges and its directional dependence be explained?
You can use the fields from the Lienard Wiechert potentials to calculate the Lorentz force $\vec F = q \ (\vec E + \vec v \times \vec B)$ on each charge.
If you do this there will be a repulsive electric force between the charges and a magnetic force. The magnetic force will be attractive when they are moving parallel and repulsive when they are moving antiparallel.
I would like to understand this in terms of dynamical effects of the electric field and not in terms of a magnetic field which is the common approach for correctly treating and not further looking at such details.
You can use the Lorentz transform to boost to a reference frame where $\vec v=0$ and then use the Lienard Wiechert potentials to calculate the E field and then calculate $\vec F = q \ (\vec E)$ for the charge at rest. Then repeat for the other charge. It is probably a little more math but it certainly can be done that way if you want.
When you do that the E field in the boosted frame will either be increased or decreased so that the E field force matches the total force that was calculated above.