I've found a couple of old topo maps of the area:
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130.166.124.2]
Based on this, I've determined that I was wrong about this station being the Soldier's Home Station on the branch from the Santa Monica Airline. That station would have been about 1/4 mile to the east of Pershing and Dewey.
Instead, the 1925 map shows a branch coming off the Westgate (San Vicente) Line along present day Bringham Ave and then turning northeast along present day Pershing to the vicinity of Pershing and Dewey. I have not been able to find any other mention of this branch (although it is stated that much of the passenger traffic and the only freight on the Westgate line came from Soldier's Home).
The 1934 map does NOT show this branch so it was apparently abandoned between 1925 and 1934.
There is no indication of a rail connection between this station and the Santa Monica Airline branch (which saw passenger service until 1953).