Some other options per Wikipedia...
List of secondary and special-issue World War II infantry weapons
United States
High Standard HDM pistol
M1941 Johnson rifle
M1941 Johnson machine gun
M42 United Defense submachine gun
M50 Reising submachine gun
M97 Shotgun (Used in the Pacific War)
M12 Shotgun (Used in the Pacific War)
Ithaca 37 (Used in the Pacific War)
Browning Auto-5 (Used in the Pacific War)
Of those, why not a High Standard pistol, maybe even pay the tax stamp and create a recreation of an HDM from a basic HD pistol?
I am not real sure about the Browning Auto 5?
Browning would later license the design to Remington, who produced it as their Model 11 (1905–1948). The Remington Model 11 was the first auto-loading shotgun made in the USA. Savage Arms also licensed the design from Browning and produced it as their model 720 from 1930 to 1949.
The Remington Model 11 Riot Gun and the Savage Model 720 Riot Gun were both very popular WWII shotguns and while both guns are clearly the Browning design (that unmistakable Browning Auto 5 look) I can't find any reference to an actual Browning Auto 5 being used by US forces?
I see the following shotguns used by US Forces during WWII:
Winchester
Model 97 trench guns, riot guns and training guns.
Model 12 trench guns, riot guns and training guns.
Stevens
Model 520-30 trench guns, riot guns and training guns.
Model 620A trench guns, riot guns and training guns.
Ithaca
Model 37 trench guns, riot guns and training guns.
Interesting note is that Ithaca completed its military shotgun contracts by the end of 1942 to concentrate on manufacturing the M1911A1 .45 pistol. All in all Ithaca only produced 1,420 Model 37 shotguns making them very rare.
Remington
Model 11 and Sportsman riot guns and training guns.
Model 31 riot guns and training guns.
Savage
Model 720 riot guns and training guns.
I can scan some images if anyone has an interest in any of these shotguns.
Ron