Electrical – How to handle the grounded neutral for a simple switch

You should run a new 14/3 cable. Do not run a single wire.

You'll never be able to fish box to box with the boxes in the wall. If the distance is close enough you might get away with only removing one box, but most likely you will need to remove both boxes and replace them when you are finished.

Code is a guide for localities to establish rules. As such, if a locale adopts the code as is, it will be up to them to enforce their rules. Now we are down to the inspector's decision as to whether you comply.

Since the switched outlet is serving functionally as a luminaire, many inspectors would probably take the view that luminaire switches need a neutral (grounded) conductor. Usually when there are two rules and one is permissive and one is restrictive, the restrictive one will apply (unless the permissive explicitly waives the restrictive).

Further, there are practical reasons to include the neutral at a switch for an outlet. Advanced switches, such as remote control devices or a proximity sensor, usually need a neutral. Obviously, devices that would compromise the outlet if used for purposes other than luminaires should be avoided. You don't want a switched outlet with a dimmer to be accidentally used for a vacuum cleaner.

Bottom line, the local inspector will probably be the last word, but I would include a neutral.