Pulling outlet from existing 220 switch
Can't put a receptacle here

Receptacles are not allowed on a circuit if more than half its capacity is provisioned to hardwired loads.  A built-in air conditioner circuit will be 22A or so.  

Can't put 15/20A receptacles on a 30A circuit

The circuit is breakered for 30A.  That means you can't have a 15A or 20A recep on it, in any case.   

The wall-wart doesn't lend itself to hardwiring... Unless it does

All the Dyson power supplies I turned up in a search were "hang off a socket" type wall-warts, and had a built-in plug matched to local power.  Presumably yours is NEMA 1-15 or 5-15.  If instead of a plug there were a pair of wires, we could make a case for "hard-wiring" the load, dodging the "no receptacles" and "no 15/20A receptacles" rules.  However, I don't see a way to attach wires to that which I could defend to the inspector with a straight face.  

There's also a question of whether the power supply is a factor in the battery charging.  Batteries - especially the popular lithium batteries - absolutely require circuitry to prevent overcharging.   If the vacuum simply wants a generic DC voltage such as 12 volts, and has that charge-controller circuitry onboard, then we can substitute any generic DC power supply of the coarsely correct voltage, and that can include supplies that lend themselves to hard-wiring.  However, if the "keep the lithium battery from exploding" charge control circuitry is in the Dyson wall-wart, then we ned a Dyson branded supply involved somehow.  

See if Dyson makes a car charger (cigarette lighter) for it.  If they do, then any 12 volt DC power supply will do that supports hardwiring. It can be brought out to a cigarette lighter socket if it doesn't have one built-in. Then, use the Dyson car charger.  

Don't trust that neutral

It's probably not even connected in the panel. By the way, the way you test it is measuring 120V between each hot and neutral.  Regardless, I would just wire up the load 240V, as that dodges multi-wire branch circuit complications.  Do not under any circumstance put a NEMA 5 socket on 240V.