Do I HAVE to have a PhD to get a post-doc position?
postdoc without doc cannot exist... You might be hired as something else, another description, same job, but not officially as a post-doc, by definition.

As commented, the funding might just not be there for such different position. I'm sure that, if I'm reviewing spending information from a grant, I wouldn't agree with that switch, unless it is significantly cheaper. Further, most of the sponsoring agencies I know, have a some sort of guidelines for paying people, and it would be difficult to justify someone without a phd receiving postdoc level money...

Your question sounds like "Do I have to go to medicine school to be a surgeon" or "Do I have to pass the bar to practice law". Doesn't matter that you think you are qualified, officially, you are not.