Warning: May become very grindy
Go directly to Gremlin cave then go directly to TotHK leaving climbing set in wilderness. You may not ascend until the ring has been retrieved. That means no use of '<' on any level.
Retrieve the climbing set from library and Axe of the Minotaur Emperor whichever order you prefer.
You may not cross the pass until you have both the axe and climbing set. Challenge complete.
Edit - Try to avoid fighting in wilderness, you may fight if there is no other option. Edit 3. Definately no picking up of items or eating off the ground during wilderness encounters.
Edit2 - Warning: Extremely hardcore, you have been warned.
Edit3 - If you want to try a lighter version - you may move between levels prior to ring retrieval.
Also a very light version - You may also fight and collect in the wilderness en route to dungeons.
Also extremely light version - You may also do whatever to the northwest of the pass prior retrieving the ring, even walk around the wilderness and fight for all eternity.
In all versions you must drop lose the climbing set after crossing the pass and may only cross the pass after the challenge is complete.