1934-1940
5505 Melrose Avenue, Los Angeles, California
Recordings Incorporated Studios, 5505 Melrose Avenue, Los Angeles, California
Date: 8/8/34
Before 1940, the studio and transcription business located at 5505 Melrose Avenue was known as Recordings, Inc. and was used extensively by Decca from the very beginning of its recording history in 1934.
When Recordings, Inc. went out of business in 1940, Decca took over the building at 5505 Melrose Avenue, installed all new equipment,and became Decca Studios, Hollywood.
September 1944
Decca Recording Studio, 5505 Melrose Ave., Hollywood, CA
http://stevenlewis.info/crosby/1930s.htm
1934
Aug 8 Records Decca's first songs at Decca Studios at 5505 Melrose Avenue, Hollywood, 'I Love You Truly,' 'Let Me Call You Sweetheart,' and two others, under contracts that eventually run to December 31, 1955.
Jul 5 Bing records Love in Bloom and three other songs in Los Angeles to conclude his contract with Brunswick Records.
1936
Jul 24/29 Records songs from Pennies From Heaven in Hollywood with Georgie Stoll and his orchestra.
Decca California Pressing Plants
960 N. La Brea Ave
Built in 1933. From 1945 through 1954 the build was the home of Decca Records pressing plant.
https://mediadistrict.org/history/
My guess would be that L.A. stands for Los Angeles. Decca's recordings in New York don't have any prefix, but the Los Angeles recordings usually have DLA or, as here, L.A.
The matrix number of "Black Moonlight", LA. 4, is the same on both releases, Brunswick 01596 and Columbia DB 2085, so yes, they are both the same recording from 1933.