electromagnetism - Force between two point charges moving parallel to each other

When we observe two point charges moving parallel to each other we can see two forces acting on each of the charges:

• the Coulomb force


• the magnetic force ($\mathbf{F}=q\mathbf{v}×\mathbf{B}$) (similar to the force between two parallel wires with the same current)

However, if we change to the charges' frame of reference, there will only be one force - the Coulomb one, and moreover, the amplitude of this force should be the same as the previous one's (the distance between the charges is constant and the force is independent of velocity). The resulting force should be independent of the frame of reference, yet it isn't and I can't find the missing piece. I know that a magnetic field is just an electric field viewed from a different frame of reference, but it doesn't help in this specific case.

I found this question to be similar: Two electron beams exert different forces on each other depending on frame of reference?, but in one charge scenario the charge density has no meaning.