Does rest exist?

All your reasoning is correct, except for the conclusion!

You take your thought experiment to mean that absolute rest exists. But what it really says it that absolute electric and magnetic fields don't exist: observers can disagree on what they are, just as they disagree on space and time. The real, invariant object is the electromagnetic field tensor $F$. Its components $F^{\mu\nu}$ in a particular frame tell you the electric and magnetic fields in that frame.

This is still consistent with your observation that there's a magnetic force in one frame and no magnetic force in the other, because we never measure magnetic force. We can only measure the total electromagnetic force $f$, given by the Lorentz force law $f^\mu = qv_\nu F^{\mu\nu}$.