A steady, straight current that is stationary can exert a Lorentz force on a moving charge that is parallel to that current.
Conversely, a stationary charge can experience a Lorentz force that is "parallel" to a moving wire of straight, steady current, though the wire may appear distorted due to its relative motion and extended nature.
If we suppose the distortion due to non-simultaneity could be disregarded as negligible, then, we could say that our charge can, in all practicality, experience an electric force parallel to that wire, according to its own rest frame. The total Lorentz force on the charge scales roughly linearly with small changes in the relative velocity between the wire and the charge, so the distortion simply needs to vary at higher powers of velocity for us to discard it as irrelevant. Therefore, in the rest frame of a charge, a moving wire clearly is able to exert an electric field on it that is parallel to the wire.
That being said, two uniformly moving line charges can be combined to produce the steady straight current. Individually, the relativistic corrections to the electric field of each varies roughly with the square of their observed velocities. The difference between the two fields varies linearly with small differences in the relative velocities of each line charge (i.e. drift velocity) and linearly also with respect to small changes in their average velocity with respect to the observer (motionally-induced emf).
Therefore, it appears quite straightforward that the electric field of at least one of the two (implied) line charges comprising the electric current has components which may be responsible for electric forces exerted parallel to the wire current, depending on the perpendicular motion of the wire. This implies that the electric fields from that moving line charge are not perpendicular to it, but rather come off at an angle at the leading and trailing sides of that perpendicularly-moving wire.
It should be possible to choose a different frame where it is the positive charge, not the negative charge, that is responsible for the electric forces parallel to the moving wire. That simply requires a Lorentz boost in parallel to the wire. This is not problematic since it simply confirms the idea that the electric force should not vary in the direction of a Lorentz boost. However, it becomes problematic as soon as our wire has a significant net charge, since then it would be impossible for the skewing of the positive line charge's electric field to exactly cancel that of the negative line charge.
Of course, we can argue, as some have done, that the electric lines of a steady moving line charge do not skew, but rather stay perpendicular to that line charge. Admittedly, most of the arguments consider only motion of the line charge parallel to its... line. But surely it has been implied in the literature that, without further qualification, that the field lines of a moving line charge cannot not skew. Who argues for the case that the field lines cannot skew must however then explain why a moving wire current, modeled as the linear superposition of two line charges, be able exert electric fields with components parallel to that wire if the fields of each line charge cannot skew, like this:
That means we have to choose between one of the two following statements:
1) A moving line of electric current can be modelled as the linear superposition of two line charges.
2) The electric fields should not vary in the direction of a Lorentz boost, and the electric field lines of a line charge with skewed uniform motion should not be skewed.
It means if one of the above numbered items is true, the other must be false.
Which one is false?