I’m looking for the beginnings of bejesus, as in “You scared the bejesus out of me!”.
Also, do the rest of you pronounce this word as Ba-Geezus or Bee-Geezus?
According to the OED, is comes from “by Jesus.” Dates from 1908 as an interjection (“Faith and bejasus!”).
I’d assume the idiom “beat the bejesus” means to beat someone until he cries “bejesus.”
The the origination part of your question has been answered (at least, I know I can’t do better than a cite to the OED), I’ll help with the second part:
I pronounce it somewhere in between the two options you posted, it’s more of a “buh-geezus”. Or maybe “Bih-geezus”.
And now you’ve had me sitting at my desk muttering “buh-geezus…bih-geezus…buh-geezus…bih-geezus” under my breath. Way to go. 
Be-Jeezus is my preferred spelling. I pronounce it "Bee-jeezus.
Based on my child hood, and close association with a Parent and some GrandParents who were heavy of hand, I would suggest that it originated among the Irish who,intent on getting the bejusus out of ireland, brought it over on every boat they could buy hold space on.
My ears still ring with "I’ll beat the bejesus [or bejaysuz] outa ya!
EZ