Hi
As I understand your symptoms, the brake plate is in an incorrect position when referencing the brake torque arm, (or vice versa), - this cannot be affected by how the wheel is laced. You need to investigate the torque arm and how it is attached, is it the correct part, is it distorted, can the fit be improved by swapping the position of the arm relative to the mudguard bridge?
How the wheel is built will effect the offset and thus the wheel alignment when installed - pulling the wheel across by 5mm is easy if the rim is concentric and aligned. Rig up a jig to hold the spindle and a pointer to gauge the position of the rim then loosen the spoke nipples one half turn on the side you want to move the rim from then tighten the spokes on the other side by 1/2 turn, check to see how much the rim has moved, repeat as necessary. When the rim is where you want it you need to confirm there are no spokes protruding past the nipples which could puncture the tube, quickest way is to remove one of the tyre beads and run your fingers over the nipple ends, any protrusions and the tyre will have to come off to allow the spokes(s) to be ground back.
I'm of the same opinion as Clive, the end caps are the same on both sides
Regards Mick