Regan is an Irish surname in my family that moved on to being a nickname for my uncle, and we pronounce it as "REE-gan". I have no doubt that at one point the surname was pronounced RAY-gan but they must have gotten tired of correcting people. While I prefer Ray-gan, Ree-gan isn't bad, and most of us have moved on from The Exorcist (I only knew about the connotation when I asked about
Regan in this forum a few years ago myself). Reegan isn't very nice looking though. I would suggest using
Regan on its own.
“Fairy tales are more than true; not because they tell us that dragons exist, but because they tell us that dragons can be beaten.”
― Neil Gaiman