Dont know about AE, but i did find another source of WW2 music. A folk musicain named Oscar Brand, collected a bunch of period and unit music during and after WW2. Note that a fair amount of his stuff is PG13. A lot of it is out of print but check on Amazon for some of it.
Four Albums of Military Song from Oscar Brand [BOX SET]
Some of his songs are :
Save A Fighter Pilot's Ass, Army Air Force Heaven, Glory Flying Regulations, Destroyer Life , To A Wave , Guantanamo Bay , The Reuben James , Steam Torpedo, The Battle Of Ormoc Bay, Laws Of The Navy , Sub Division Nine , We Aint' Going To Sea No More , The Reserves' Lament, Wake Island , Man Behind The Armor-Plated Desk, Paratrooper Song , Quartermaster Corps , Infiltration Course