After thinking about it for a few minutes, I don't think I'd even consider trying to laminate a dough as stiff as yours, although I think your dough is probably very similar to fazzari's dough (at his pizzeria). Interestingly, though, fazzari never uses dough with that low of a hydration figure with pizzas that he makes at home (using a rolling pin). I'm thinking he uses at least 40% hydration at home, specifically because dough that's any stiffer is just nuts to laminate. I'd think it's possible to roll your 36% hydration dough by hand as long as you're not laminating it. If you really want to laminate it, though, I'd suggest increasing the hydration to 40-42%.
Also, I'd recommend peeling the topped skin onto a stone, rather than baking on a pan (if you currently use a pan). Prior to this past summer, I baked laminated crusts on a perforated pan, but I stopped using the pan over the summer, and the resulting pizzas are much better.