OK, time for a Sturmey primer. I started to write this but finally found the same thing with some creative searching.

"In low gear, the clutch is moved to the right to lift the (high gear) pawls, driving the ring gear directly so that the ring gear drives the planets with output going through the planet cage and its (low gear) pawls at 3/4 the input speed. With one pair of pawls raised and the other pair driving, the typical clicking sound of AW hubs is absent.

In second gear, (direct drive), the clutch drives the hub body directly through the right hand pawls, making the planet carrier (low gear) pawls click as they lag behind.

Top gear engages when the clutch is fully extended to the left against the planet cage, between the four protruding planet (pinion) pins, to drive the planets that turn the ring gear 4/3 the input speed. In this position, the low gear pawls in the planet cage click as they lag behind. It is the inverse of low gear and hence the reciprocal relationship between low 1/3 down and high 1/4 up with respect to direct drive on AW hubs."

As you can see it's not possible to "go to 3rd on the way to 2nd" nor can 3rd be related to the tension on the cable. Because 1st and 3rd use exactly the same transmission path, but in reverse to each other, the gears and pawls themselves are all intact and working properly. So the cause has to be that the clutch is not able to push against the protruding planet pins (pivots). Occasionally it is necessary to backpedal slightly to engage 3rd but I doubt with all the effort the OP has made that it's that simple.