Pardon me if it seems like I am taking the wind out of your sails, but I think you are approaching this backwards re: your search for information on Bernard E Bay.
The citation regarding the Hawaiian Department is NOT absolute.That was only a pool of people being taken in for a stated purpose.That is reflected in the micro second it took to create his enlistment card.What happened to him, enlistment-wise, five minutes after the card was created is another thing entirely.My advice: Forget about the Hawaiian Department and the division roster search....Find him as an individual back in Ohio and search from there.If it eventually leads you to Hawaii and the SPWA that's fine....but to begin with Divisional Rosters and work backwards is going to take an eternity.
BTW--I seriously doubt that an August enlistment would put him in Peral Harbor in December.Guys were still taking long term basic at that time so four months would not even have him out of state-side training yet.
I see you pasted the enlistment doc from NARA---there is a typo--there is no such County in modern Ohio as: MUSHINGUM---it is Muskingum---county seat Zanesville.Bernard is written up in the 1930 Census as BARNARD...and it is a HUGE family....have you found any of them yet?
I suggest you get into the Zanesville Historical Society and/or County Library and start looking for citations for Bernard.