To the OP,

Although I agree that you should have Flagship access in LAX, be prepared for potential disappointment.

It is simply that the rules are so very complex and you may come across someone governing entrance to a lounge that is ill-informed. Last week I went through MIA and LAX (heading towards Australia) and was granted Flagship access, but it was made clear to me that this was due to Emerald status (not through flying business).

My experience over the years have made me realize that nothing is sure. I truly hope that you get the access, as it is far superior to the vanilla Admirals experience. But be prepared for the chance that it may not happen.

Having said that, when you do get in to the Flagship lounge, this doubt will make that even better

Juddles.