I grew up near Principia College, a Christian Science liberal arts school with a beautiful campus.
I once interviewed at Salve Regina University and when I got there I of course pronounced it as a Latin name--A as in Ave Maria, V as in veni vidi vici and G as in Gaius Maximus--since, after all, it is a Latin name. I was corrected very quickly. They pronounce it locally in a very anglicized fashion. A as in apple, v as in very, and g as in German.
Sure, you can pronounce Principia correctly in Latin, and there's probably nothing wrong with that, especially if you're speaking Latin, but if the word appears in a proper noun, then you're probably better off pronouncing it as the locals do. I voted, before reading the OP, as in the title of Newton's famous treatise on physics (option 3), but I could also vote in some other way depending upon the use of the word as a proper noun. FWIW, I've never heard the word pronounced as in option 2. I know physicists who anglicize it (option 1) and those who pronounce it more conservatively, as I do, but never heard the option 2 version. Certainly I will defer to the local population in the pronunciation of the university.
It's Like Medina, Ohio and Etoile, Texas, and Winona, Minnesota. None of these cities are pronounced "correctly" by the locals, but the local pronunciation is the right one if you want to be taken seriously by the people you meet when you visit these places.
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