My proposal:
Use a heavy duty and high quality, say 5000w, transformer to convert the 120v(from the outlet I will be using for both rigs) to 240v. (Other small appliances are unchanged)
This way, that outlet should be capable provide 20A*240V=4800w of power and then the needs of rig 1&2 are satisfied, plus the PSU will operates more efficiently at 240V.
Will this work?
No. While your "transformer" will be delivering 20 amps at 240v, it will be sucking 40 amps at 120 volts from the wall to do it, thus overloading the circuit. Watts in = Watts out. Assuming your transformer is 90% efficient, if it was sucking 40 amps, it will only be delivering 36 amps + heat. Also, I do not think a transformer that size is under $200.
Note: 120v is not really 120 volt. It is really 110 - 120 volts, and could be lower during brownouts. So, if you want continuous load, you should be thinking 110v x amps, or even 110v x amps if your area has bad power. If so, you should be using power conditioners to protect your rigs, and that increases the power draw at the socket.
One possibility, but only if this is your house, since it will require electrical work:
You could convert the 110 to a 220 volt outlet. If you are not too far away from the electrical box, you could run a second hot line, and connect it into the other phase coming into your house. Then change the wall socket from 110v to 220v. If you have conduit, it could be relatively easy.
Note: It is not that much more difficult to just run another 110 circuit. But, if there are conduit limitations (too many wires in the conduit), then only having to run one more wire could be the solution.
Another possibility is to get another 110v outlet on a new circuit near the electrical box, and run a really heavy duty extension cord.