Just to be clear on the soft start thing ... what level of peak inrush current should one expect from a toroidal transformer rated at, say, 120V primary and 750VA? If it's 10X the 6.25A then we're talking 62.5A give or take IF there's no series limiting resistance. Let's say we use a 10 ohm resistor in series with the primary then the max current would be about 12A but decline as the core sorts itself out.
So, if the outputs were disabled during this time and then, once the core was OK the input resistor is bypassed and the outputs enabled. The input resistor would not need to be rated at 1440W as the length of time the resistor would see that level of power is small and over the 10 cycles, assuming the current drops to near zero, the resistor would likely see an average of about 800W for 1/6 second or about 133 joules. Many camera flash units dump more than that energy during a flash. I would think even a 5W resistor would be enough.
So, to layout the components needed for a softstart circuit you'd want a time delay relay at the input that picks in, say, half a second, and a 5W 10 ohm resistor in the hot leg of the input to the transformer that will be bypassed when the input relay picks. The output could be controlled by a time delay relay and pick a half second after the input one picks OR the output could be turned on manually by pressing a button that picks it -- perhaps using a circuit in the input relay to inhibit turning on the output until after the input relay has picked. Sort of a modified EMO circuit...
Brian