" I have looked through the map axis properties but couldn't find anything allowing me to change the label position relative to the axis. ...Am I missing something?"
I don't have the toolbox to be able to test, but perhaps.
Try
saxMap=struct(axMap)
which will reveal all properties of the map axis object including those that are normally hidden. Oftentimes one can find the pieces of the object looking for that TMW didn't think were important for the end user.
Good luck "handle-diving"! :)