and your not IMMEDIATELY and accurately on the rudder, the localizer shots out to the side and it turns into a go around.
That is entirely expected. When close to 200 ft which is a typical Cat One DH or DA (whatever, depending!) the Localiser beam is getting progressively more narrow and seemingly more sensitive. Depending on the runway length among other things the localiser would be around 100 metres wide from edge to edge at the landing runway threshold. Note: Purely a guess but I am sure some maths guru will come up with an accurate figure. You would only have to yaw a few degrees during the intial one-engine inoperative go around manoeuvre to slide outside the localiser width near the threshold and surely only a pedant would fail you for not maintaining the localiser centreline so close to the threshold durng the climb.