Hi Tony,As I've recently learned from @bertus, Japan has a 100 VAC system. At this point I would not suggest trying to plug your supply into a 240 line. Not unless you don't mind chancing the possibility of blowing out your supply.
Back to your original proposal: Use of a Variac (variable transformer) - I don't see why it would be a problem. But then again, it's not my money and equipment on the line. At this point I can give you no further assistance. I hope you solve your problem. ME? I'd opt for buying a supply that can work from a 100 to 240 input.
You don't say where you're from. I'm in the US. When I want a 12 volt supply of around 2 to 4 amps I just go to the local cable company and ask them if I can have one of their returns. They don't reuse the old supplies, and I often can get two at a time. Xfinity and DirecTV are a few places I can go to get power supplies for free. If you have such in your area, perhaps you can give that a try. I also find a lot of old printers that have external power supplies. I've found one at 30 VDC, one at 18 VDC and one with two different voltages, I don't recall. I don't know if that will address your needs but it's worth a shot if you can get your hands on something like that. Last year I modified one of those freebee PS's from 12.1 volts to 13.8 volts by varying the resistors on the reference voltage. I use it to keep an old car battery in the garage topped off so that when I'm working on the car I can play an old car radio. Works sweet.
Bottom line, be creative. But also be careful.
Thanks for your feedback. I will be taking the risk and plug it into my Variac.
JL