Craig, great photos. I like the captions
I'm encouraged that your oven can reach 570, but, I hate to say it, from the dark color of the stone, I don't think your soapstone tile is the kind of soapstone that's ideal for baking. The whiter the stone, the more talc it contains, the better it is for baking. The fact that it's taking so long to reach 570 is another sign that it could be a poor conductor as well.
You can certainly try it and see how fast of a bake time you can get out of it, but I'm not all that hopeful for this tile. Don't pull the trigger yet, but I'd do some preliminary shopping for steel.
Hearth materials are about conductivity and thermal mass. The conductivity is the rate at which they transfer heat and the thermal mass is the amount of heat that they can store. At the same temperature, two materials of different conductivities will bake the pizza at different rates. High conductivity materials (like steel, and to a lesser extent, soapstone) are a way of making a 550 oven bake a pizza in the same time frame as a 625 oven or even a 650. These materials let you bake fast pizza without messing with your oven.
You can definitely get Paul's pizza out of an LBE, but it takes a lot of tweaking to dial in the deflection right. I think your initial inclination towards steel is a good one. Bake a pie or two with the soapstone, but keep steel in the back of your mind if that can't break a 5 minute bake time with a two hour pre-heat.
While not perfectly ideal, KABF will work pretty well for this pizza. Paul, being in the UK, is using an unbromated flour which is most likely very similar to KABF. Until you can deal with a distributor, stick to that.