Linwood, King Arthur Sir Lancelot (KASL), is not ideal for thin crust pizza (better for bagels). Although it's in a similar realm protein-wise as All Trumps, the All Trumps is bromated and bromate is a volume enhancer, making it easier to get puffy crusts.
The All Trumps has a little too much protein, though, so it's best to blend it with a lower protein flour- not the Caputo, but something like all purpose. Ideally, for this style, you should be looking for a mid range protein bromated flour like King Midas Special, Full Strength or Spring King, but those tend to be a lot harder to find than All Trumps.
Just about any flour can make a dough that's easily stretchable. With a nice long proof after balling, you should have no problem stretching the dough. In fact, for a thin crust pizza, you really shouldn't be doing any kind of bulk ferment. Just make the dough, knead it, ball it, let it ferment until doubled in volume (ideally overnight in the fridge), let it warm up a bit and then stretch and bake.
An All Trumps/All Purpose blend gives you a tiny bit more extensibility, but, more importantly it gives you better volume and a better crumb.
I'm curious, what kind of stone are you using that gives you a 4 minute bake at 500 degrees? Are you par-baking the crust? Does your oven actually run at 500 or does it have a tendency to run a lot hotter?