Mean Well power supplies like the one you linked have trimpots next to the output terminals to adjust the voltage. The spec sheet says that the SE-350-12 can be cranked up to 13.5V, which may make a difference. The big question is how your RAMPS board will handle the higher voltages. I don't use one, but have heard that there's concerns if your voltage exceeds +12.00001 volts. If you can get a stable +12V out of your ATX power supply, you could run the +12V to your RAMPS from that but feed power to the heatbed from the Mean Well SE-350-12 through your relay with the voltage cranked up. (Be careful with the trimpot, they are fragile.)
My SeeMeCNC Onyx 300mm diameter heatbed has about the same power draw/curve as the MakerFarm one you linked above. When I was running it from 12V, I had to throw a magazine over the glass and wait 30-40 minutes while it struggled to reach 110C. I bought a Mean Well SE-600-24 (24V, 600W) power supply, and turned it down to 19.5V on the trimpot. I was already using an external SSR and heat sink, so this power supply drives the bed at 19.5V. To power my 32-bit controller board, I picked up a 24V->12V buck converter from Amazon:
http://www.amazon.com/gp/product/B00BWKXTUU/
The net result is that my heated bed is running from 19.5V, the controller is being fed a regulated 12V with enough overhead to run triple extruders if I wanted to, and I only need one power plug to a wall outlet. Granted, my wallet is $120 lighter, but you can't put a price on a heatbed that goes from 20C->110C in 6 minutes without insulation and has no problem holding (or even overshooting) 110C.....
My SeeMeCNC Onyx 300mm diameter heatbed has about the same power draw/curve as the MakerFarm one you linked above. When I was running it from 12V, I had to throw a magazine over the glass and wait 30-40 minutes while it struggled to reach 110C. I bought a Mean Well SE-600-24 (24V, 600W) power supply, and turned it down to 19.5V on the trimpot. I was already using an external SSR and heat sink, so this power supply drives the bed at 19.5V. To power my 32-bit controller board, I picked up a 24V->12V buck converter from Amazon:
http://www.amazon.com/gp/product/B00BWKXTUU/
The net result is that my heated bed is running from 19.5V, the controller is being fed a regulated 12V with enough overhead to run triple extruders if I wanted to, and I only need one power plug to a wall outlet. Granted, my wallet is $120 lighter, but you can't put a price on a heatbed that goes from 20C->110C in 6 minutes without insulation and has no problem holding (or even overshooting) 110C.....