Thanks for all of the comments and helpful advice. Much of this is still over my head but I read every comment, and already asked an electrician if he could upgrade a 120v outlet to 240v, even at the same amperage or going up to 15a. He said it wasn't possible. I may also get some more opinions on this. Going to 15a, 240v would be a big improvement over 120v@12a.
Either you need to remove the "or going up to 15a" above, or you have an existing circuit that is only 10 amps...? Those are common enough in old houses, but I didn't think they were at all common in newer ones. What you want is to convert an existing 120 volt 15A (hopefully!) circuit to a 240 volt 15A. This is very do-able. I suspect that what is stopping the electrician is the fact that the conversion eliminates to Neutral wire, by converting it to a second Hot wire. This isn't allowed for normal circuits, but it IS allowed for an EVSE (charging station) circuit.
So all he has to do is either hardwire the EVSE to the existing outlet, making sure that any other existing outlets on the circuit are removed and the blank covers labeled "240 volt 15A - NO GROUND" or install a 240 volt 15A air conditioner outlet, and make sure that that one is labeled correctly as well. From your perspective the only difference would be whether or not you could unplug the EVSE to use as a portable, or not.