Getting closer, yeesh.
So I noticed on the tab where I made the new skill Fighting linked to (Strength), there is a box: Replaces Thing ID. So in that box I typed skFighting. Lo and behold, that actually works. If a player selects this version of Fighting, it replaces the original correctly and does all calculations correctly.
Now the downside, I also have a skill Fighting linked to (Smarts). If I try and do the same thing I get an error. Apparently only one skill can 'replace' another using this mechanism.
But heck, I got one working