Been there, trial and error, but have been making pies on the grill for years and the way I do it works great...For the best pizza on the grill you need the 600 or 700 degree heat inside a gas grill with the grill lid closed or it won't cook right, but as you correctly stated with the burner flame being directly under the stone the bottom of the pie will be charcoal black before the cheese/toppings on top is done. Two things you need to do to get that corrected and the direct heat from the burners deflected and reflected away from the bottom of the stone ...#1) aluminum foil....#2) a 2nd pizza stone. First lay down a sheet of aluminum foil directly on the area of the grill grate (and ONLY on the area of the grill grate that the stone covers) that pizza stone #1 will cover. Next, get a 2nd pizza stone that preferably has legs (some do have short stubby ceramic legs) and put it directly on top of the first stone. What's important here is an inch or so of air space between the two stones. The foil on the very bottom will reflect the direct radiant heat off the bottom stone and the air space between the two stones will act to somewhat insulate the top stone from getting too hot. This set up also helps convective heat transfer up the front, back and sides of the grate that isn't covered by anything and into the top of the grill hood which is over the cheese and toppings. For pizzas to come out right you need fast cooking times (10 minutes) with high heat (700 degrees), lowering the heat (and lengthening the cooking time) to keep the bottom from turning to charcoal will only give you a soggy, doughy pie that will taste like a toasted cheese sandwich. Use the foil and the 2nd stone and you won't have to turn the heat down, your pies will be better than Pepe's...well maybe not.
My favorite pizza place BTW is Zuppardis, the best sausage and mushroom pie in the state IMO, Modern a close 2nd. Have been going to Pepe's for many, many years and it definitely has gone down a notch last 10 years or so. Same thing for Sally's, and not surprised it's for sale.
