I was wrong about the wire, it's not shielded. It's just plain Jane 12/2, so that's good.
Here's the core I started with. It's 97 lbs.
I don't have any way of disassembling the core to make winding easier so my only option is to tediously wind it through the core window. I'm just using a bobbin kind of deal.
I've wound about 90 turns onto one side of the core so far. Basically 5 'layers', the layers got a little indistinct after a while so I use the term loosely

After winding on one side I hooked it up to 120 V and measured the idle amps at 0.7. Much lower than I thought. I was expecting at least 3. I still need to wind about the same amount of wire onto the other side then connect them in series. Then I will connect 240 volts to this to make sure things are still OK. If they are (i.e. I survive) I will wind a few turns of wire by themselves on, turn it on, see what the volts per turn are, then figure out how many turns I need to add to draw off 360 volts. Then bingo, a 240 volt primary 360 volt secondary autotransformer.
My only real fear here is that I didn't put enough gorilla tape over the laminations before starting to wind and that they might cut into the wires.
BTW Nehog, since you're an EE I have a question about capacitors. In my very limited experience of capacitors it seems that when fed AC they charge up to the peak voltage and not the RMS voltage. When capacitors are rated then are they rated for RMS voltage or peak voltage? I'm wondering if I put 360 volts AC into the 440 volt capacitors I already ordered if they will blow up because 360 RMS equates to 509 peak volts. Thanks.