Velocity composition effect of moving line charges acting on a moving charge - By what velocity (boost) is the E-field unchanged along the boost?

Claim 1) An infinitely long line current can be modeled as the linear superposition of two infinitely long line charges.

Claim 2) An infinitely long line current can exert forces on charges with components parallel to the line it's on.

Claim 3) Because of Claims 1 and 2, at least one of the line charges comprising the line current is responsible for forces parallel to the wire acting on external charge.

Claim 4) We can use two parallel, infinitely long line currents, each composed of two parallel superimposed line charges, where the net force acting on an external charge positioned in between may be parallel to both line currents. We will assume, until Claim 8, that these infinitely long line currents are moving closer to together under their mutual magnetic attraction. These wires will, of course, induce EMFs on each other.

Claim 5) Any difference to the initial amount of current in the parallel wires of Claim 4 will cause a difference in the initial amount of force on the external charge of Claim 4 directed parallel to the wire. This is expected because EMFs are induced - it's simply a consequence of Faraday's law.

Claim 6) In the rest frame of the external charge, the parallel force acting on it will depend on the amount of current flowing in the wires (per claim 5), and therefore it depends on the relative parallel velocity of (-) line charges with respect to the (+) line charges.

Claim 7) Due to linear superposition, changing the parallel velocity of the (-) line charges is sufficient to change the parallel force acting on the charge positioned as per Claim 4. Similarly, changing the parallel velocity of the (+) line charges is also sufficient to change the parallel force. Therefore, each set of line charges can independently contribute to changes in the parallel force. By doing this, one could, for obvious reasons, obtain a parallel force of zero, simply by making the drift velocity zero, eliminating the current flow, and therefore also the EMF, entirely.

Claim 8) If we considered two parallel line currents consisting merely of two parallel 'naked' (-) moving line charges, then, as per claim 7, changing its parallel velocity (while also having outward perpendicular velocities) will contribute to a change in the parallel force (due to a projection of the altered E-field in the parallel direction). There will be no (+) moving line charge in this case to cancel this change.

Claim 9) Continuing from Claim 8, the projection of the altered E-field in the parallel direction is a consequence of the fact that original velocity of each (-) line charge was not strictly parallel to their alignment, as they were moving away from each other.

Claim 10) Changing the velocity of the external charge of Claim 4 in the parallel direction invokes a Lorentz Boost that is non-colinear with the Lorentz Boost that would take the observer to the rest frame any one of the (-) line charges of Claim 9.

Claim 11) The external charge of Claim 10 would in its new inertial frame witness a different "net rotation" per electric field contributed by each the (-) line charges which were already moving as viewed in the inertial frame the charge had prior, altering the sum of the components of the electric field from each, both of which have a projection in the direction of this (non-colinear) boost - the projections of which do not cancel.

Claim 12) To elaborate on Claim 11, the magnitude of the electric field in the direction of this boost changes, not because of the boost itself, but because of rotation due to the boost's non-colinearity with respect to the boost to each of the different rest frames of each respective (-) line charge.

Claim 13) The fact that a Lorentz boost cannot cause a change in the magnitude of the electric field in direction of the boost does not mean such a change does not occur by other means, as a rotation of the electric fields of various electric field sources (subject to different non-colinear boosts) may cause the electric field to vary in the direction of the boost (but not because of the boost!).

Question 1: At which claim # does this argument begin to fall apart, have errors, etc?

Question 2: If there is no issue with the claims above, why is it said, often without further qualification, that the component of the electric field in the direction of a Lorentz boost does not at all change in the direction of that Lorentz boost, as if rotation could not happen for the electric fields of the sources that are already seen to be moving prior to the invoking of that Lorentz boost?