I have the same problem as OP has stated in this question, but more generally; I cannot see how even if Maxwell's equations are lorentz invariant, the principle of relativity still applies.
When we simply use the laws of classical electrodynamics (before the advent of special relativity). It predicts that the moving charge experiences zero force for a frame at rest w.r.t the charge. When we switch to the lab frame, suddenly there is a magnetic force. The second frame will see the test charge collide with some object where the rest frame will not. Doesn't this break some fundamental principle of physics? And if the rest frame observes the test charge also accelerate, then we have a disagreement with Maxwell's laws. I really don't think I have any clue what's going on anymore. How do I interpret Maxwell's laws and the subsequent need for special relativity?