I am taking the BAR, because I'm not letting some chicken little carrying it and letting the whole squad get pinned down and cut to ribbons. I might not last two weeks, but when I get it, it will be my own fault or dumb luck. I would, however, trade the BAR for a Johnson light machine gun
Actually, it would be hard to choose unless a guy knows where and when he would be fighting. If I am with the Marines, dug in on the ridge between Henderson Airfield and the Japanese, then I would take the .50 caliber. If I were on Iwo Jima, then the flamethrower would be indispensable.
Marching miles in New Guinea or Burma might make the M1 Carbine's handiness and light weight a lifesaver. Dropping by parachute into Normandy, a M3 or Thompson.
Oh, and I think the 1911 might be the best sidearm in all of those situations. I am sure most Model 1917 revolvers from S&W or Colt could probably outshoot the M1911, but there are no scoring rings to speak of on an enemy soldier.