Dave I'm a udge fan of those recordings ...
I'm quiet sure he used a C-37A as microphone.
I think you are just right for LA-2A, but as I'm sure he used Quad eight console and preamps, he might have tried Quad eight comp, I don't know...
I would bet on LA2A.
Sinatra was very concerned by the sound in general, and the sound of his voice in particular. He walked away from expensive sessions just because he didn't liked the sound played back ... Leaving a very expensive bill with string sessions etc to the record company
I try to find some more about Frank recordings
malice