The best case scenario would be that you could preheat the plate to 550, then turn the broiler on and have it stay on for 90 seconds until the probe hit 585. If my calculations are correct, you'll probably need 550 on the hearth. If the extra thermal mass of the steel really delivers and you can work with 525, then you should be in good shape, as the broiler should stay on for a while with a 525 to 585 jump.
John, could you take a photo of the probe in your oven? I've spent some time with Tyler (Sum1else) going over some gentle mods to get a slight temperature bump, and, while he's still finding the one that's best for him, he needs a few more degrees than you do and he's working with a far weaker gas broiler. Shielding his probe with a foil pouch didn't seem to buy him enough degrees to keep his broiler on, but, in your case, it might give you just enough a bump to do the trick.
Could you also take a shot of your broiler element? Beyond making sure the element stays on, you're going to want to shrink the space between the broiler and the stone as much as humanly possible- 2.5" or less.
You had mentioned earlier that you were going to look into the wattage on the broiler element in the top (and bottom) oven. That will go a long way in telling us how much of top heat it can deliver.
If the broiler does end up being a bit anemic, there's workarounds we can try, like using a malted flour such as KAAP. Until then, though, I'd try starting the with the Caputo, if you can get it.