And, as a follow-up, does having the answer to this question dictate how we should say it?
He pronounced it the way any Jew of Lithuanian or Russian origin would have pronounced it back then: "Feinshtain". Being that this is not our natural pronunciation of such names, it would appear to me to be disrespectful to mimic his European pronunciation. His own children and grandchildren do not use the old pronunciation unless they are among people who still speak that way.
I'm pretty sure that almost every time I've heard rabbis mentioning R' Moshe's last name, it was with a long "e" sound. The pronunciation (of anything) in this video is by no means authoritative.
I think that we should, indeed, pronounce people's names as they'd pronounce them, to the best of our knowledge and ability. For one thing, universal pronunciation is good for clarity, and there's no better canonical pronunciation of a person's name than that person's own pronunciation. For another thing, it seems to me that pronouncing people's names as they'd pronounce them is an act of respect.