After the movie projector gets hit with a shell with McHale nearby, the next scene shows McHale with a torn uniform and dirtied up. He he were really that close to the explosion, he would be all bloodied up; probably worse.
In future episode Three Girls on an Island (1962) the opening title reads, "Somewhere in the South Pacific 1943", thus establishing a time reference for this point in the series. One of the films stolen by McHale and Gruber in this episode is Thirty Seconds Over Tokyo (1944), which wasn't released until 1944.
Near the beginning when Binghamton is walking into his office followed by McHale, you can see the shadow of the camera man on the metal storage cabinet as he backs away.