It would be appropriate to run your E7s [and E6s]with either the Lightweight unpainted stainless cars or the Heavyweights during the period immediately after WWII when the E7s were first delivered. The green and two tone green Heavyweights decorated during the 1920s for the Crescent Limited and other Southern "name trains" mostly ended in the early 1930s due to the Depression when many cars and engines were parked on "dead tracks" for lack of business.
Due to the major postwar production backup, Southern's huge orders for new stainless passenger cars from Pullman Standard, Budd, American Machine & Foundry and others were only delivered beginning 1949. The earliest stainless cars called the "silver cars" were delivered along with the E6s in 1941 for:
The Southerner[March '41], The Tennessean[May '41] and The Crescent[Dec '41]. Subsequent to Pearl Harbor on December 7, 1941 deliveries of cars and engines were halted with rare exceptions.
Southern painted the roofs of some of its stainless cars flat black but aside from lettering I have never seen evidence of green, gold and white decorated cars. Certain photo advertisements along with PR photos of "The Tennessean" appear to show a solid green smoothside light weight car in a consist but nothing concrete.
There are none shown in the Southern Railway Color guide to Freight and Passenger Equipment.