Okay, my tiny brain has put some effort into this, and I took a photo from below inside, showing just how difficult access is from the rudder-up handle area to the Harken turning block (which redirects the rudder up-line back to a "flying block" behind the rear rudder-up handle on my TI). The flying block goes back to the rudder, with a bungee tensioner (needed to retract the rudder-up handle when released).
In the photo, I have passed some solder through the handle tube to show the relationship with the turning block.
I now see that on later Islands (I have only seen the 2016 AI2, so don't know when it was introduced), the turning block has been moved towards the bow, enabling easy access from the front hatch.
So rather than attempting to "thread the needle" by pushing line through the handle hole and fiddling around in the hope that the end of the line will magically drop into the block, I will fit another block up forward like on current models. In addition, I will add a swivel between the new line and the existing one, in the hope that I won't get any twists again. Either way, I will be able to untie the line at he joint to remove any twists.
Some of you might be asking why this has happened to me? Well it might be a clue that the other Island users who mentioned having similar issues are prolific users, suggesting that the line twist occurs over time. Hobie's (and anyone else's) relocation of the block should at least simplify the fix.
I would like to thank Hobiecat Australasia for going above and beyond to help me solve this issue.