500W @12V ~45 A , D3 must be rated for this current, also it will drop .7 V so output on ATX will be 12 V - .7 = 11.3V, power in D3 ~ .7 X 45 = 35W, it needs a heatsink. Likewise D1 and D2 dissipate 35 W between them, hence must be heat sinked. I suspect your SSR will drop volts and so that will have to be heat sinked. You must specify a type number that can handle the 45 A, this will then give you its input current for switching, I believe its normally about 5 -20 mA. I think that you need to put in some hysteresis so if the ATX 12 V starts to sag, the circuit flips over and will not come back until the ATX is back to 12 V - I think this should be adjusted upwards to compensate for the D3 voltage drop, so set it for 12.7 V. Oh yes some form of indication on the ATX that its 12V is good, else the first time you know its failed is when the two batteries run out of puff!
Frank