I think that using the return wire to prove a point is cheating. It does nothing for the original scenario without a return wire.
The ladder paradox also has some asymmetries that seem to be missing in your example:
Figure 4: Scenario in the garage frame: a length contracted ladder entering and exiting the garage
Figure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder
The two frames do not see the same number of rungs inside the garage in each case.
If we assumed that the protons were represented as tiles on the garage floor, the garage as the wire, and the ladder as the electron current in and out of the wire, then clearly the charge inside the boundary of the garage is not invariant
However, considering that the electric field intensity increases by the same amount that the boundary of the garage in the LT frame is length contracted, this would keep the electric flux around that boundary of the garage a constant.