Good info Scott.
Using the formula above recommends 12% oil and 6% butter. The crust was quite soft, do you think the pizza you had had that much oil/butter?
Im not sure of the exact dough recipe at the Lynwood specifically. My gut is telling me that 18% fat is too high because it would be too soft, but I have never tried to make a pizza with that much fat in it so im not sure. You want it to be a bit biscuity but its got to be able to stand up on its own and have some crisp to it.
I have heard that the Lynwood uses butter in the dough which would be consistent with a good biscuit recipe. I have always wondered if its possible that they are just using oil or shortening for the fat since you often find that in commercial operations where cost savings is typically paramount, and that the butter flavor people think they are getting from the dough is really just the butter flavored pan release spray. I know for sure that many bar style pizzerias use milk in the dough and even some that use eggs but I don't know if that is being done at the Lynwood.
I think a good place to start on nailing down the oil amount is to figure out oil/butter/shortening amounts in a typical biscuit recipe and then cut that in half... I don't think there is as much fat in the dough at bar pizzerias as there is in a biscuit, its just going in that direction, more than say a NY pizza dough recipe thats usually in the 0-2% range.
The crust for bar pizza is not consistent throughout the various restaurants that make this style. Some lean more typical NY but with a pan rise for their crust, some lean more to a biscuity crust, and some are more crackery than others. One thing I don't taste in a lot of bar style pizzas is much if any sugar in the dough... but some do have a sweet sauce.
Its really the cheddar cheese and what the pan does to a thin crust that gives the majority of the flavor and textural profile to these pizzas. I have many times used leftover dough from other recipes, NY, focaccia, etc and pressed them into a lightly oiled 10 inch pan topped with a good crushed tomato and a good white cheddar to find my friends/neighbors/relatives who have been eating bar pizza since birth tell me that it was the best bar pizza they have ever had.... so my point here is that the dough recipe can be altered more than you would think... its not the defining characteristic to the style in my opinion.