Using Flusser's Style and the Man and the Robb Report's list of top tailors as my guide, I discovered that there are at least three bespoke tailors of note in the Los Angeles area. They are (or were):
Unfortunately Mr. Gasbarri passed away in 2003. According to a Variety obituary, Gasbarri's clients included Red Skelton and Steve McQueen. He specialized in a unusual jacket design with no center back seam.
The other two - Taylor and Trabalza - are rather advanced in age (octagenarians I believe) and tailoring institutions in their own right. I hope they continue to ply their trade and hope that they have a team in place that can continue their tailoring traditions.