Nice'n'strange. I wouldn't do too much to it for fear of making it normal.
Perhaps more bass or a more defined bottom or middle for it - but that'd be a normal thing to do.
Have a listen to some of the earliuer residents LPs - Meet The Residents or Third Reich'n'Roll. They us ethis style of vocal & use some found or toy instruments to add some menace. their later stuff uses the voice but with more conventional instrumentation - which, I guess, is closer to you so try Man's World or Booger Breath.
Oh, I like it....
I wouldn't limit it by calling it jazz though.