"PRE-rogative" is the correct spelling of the word, but "per-rogative" is the common pronunciation of the word. Like pronouncing FEB-ROO-ary as opposed to FEB-YOU-ary...noo-clee-yer vs. nuke-you-ler.
It's all personal preference.
All things are personal preference these days...like running red lights; throwing trash on the highway; etc., etc.
You can be named "Bucket" and pronounce it "bouquet"...but folks who know better still laugh at you behind your back.
I have an acquaintance whose pride in life is correcting others folks' grammar and pronunciation. It's rude and annoying and grates on most of us.
Yet, he pronounces the perfectly wonderful French work "outrč" as "out-tray." He uses the word a lot. His associates who know better (i.e., it's pronounced "ooh-tray") haven't corrected him as that, in their minds, is bad manners. I haven't corrected him because it pleases me not to.
I'd love to be there when he learns the correct pronunciation of it.