Since you're working with AWS D1.1, I'd say go with AISC. That's the typical combination for structural applications. Do you have job specifiations? If so, the requirements should be addressed there.
Joe,
I have the utmost respect for you and your input into this forum. That said, I can assure you that when AISC does a "full audit", they do indeed get out on the floor and query the welding and yard personnel, as well as auditing the procedures. I have seen them go from playing golf in the old days to currently doing a very intense audit. They have made us a much better company than we were 20 years ago, and our bottom line reflects that. Of course, some auditors are "easier" than others, but most of them are very professional and very thorough.
Dan