I use a Natural Levain, plus IDY Kicker for Oven Spring and do a 72 hr cold rise. Then Let it sit 2hrs at RT before Stretching the Skins.
Looking to get the best extensibility, with minimal shrink back and skickyness.
I've used All Trumps for naturally leavened dough, and I didn't care much for the results, particularly when compared to what I get with All Trumps and commercial yeast. With naturally fermented dough, the crust always ended up pale and tough. But the paleness surely also resulted from the absence of top heat in my grill, as well as my lack of expertise in the area of naturally fermented dough.
The formula I listed above is probably a little too stiff for your tastes, but it's right on the edge. Bumping up the hydration a couple percent would probably make a big difference in that department.
I also used to prefer not adding oil to my dough, but I have found that just a little oil makes All Trumps dough a lot better.