OK, let's eliminate some possibilities. It's not the lever or cable at all - problem happens with a completely slack cable. It's not lack of lubrication somewhere in the system. If lubrication in general was the issue the OP would have problems shifting to or using 1st gear. The hub uses the same chain of parts in 3rd as in 1st - that's why the gear ratios are the inverse of each other.

The problem is that the clutch is not going as far in on the hub as it should. As usual FBinNY is accurate about what to try. Once the clutch reaches the proper position for 3rd that gear will work properly unless the gears are damaged, in which case I believe he would still not get direct drive. It's been about 30 years since I overhauled a Sturmey 3sp but I did dozens upon dozens and I would say the problem is indeed something interfering with the clutch reaching the 3rd position, as FBinNY noted.

It's not really that hard to disasemble and service a 3 speed hub - I really enjoyed it. Sheldon give the procedure very well. If parts are damaged you will most likely need to scavenge. Your title says AW hub but your post says AM. I hope the latter is a typo, as if you truly have an AM hub (narrow ratio) that could be more extremely challenging - it's very rare. In that case if the bike has sentimental value you might be better to replace the entire hub.