He says he had a permit stamp on his passport.
How do you know they didn’t have such a stamp?
Because I know as a fact that one didn’t need permission to leave Cuba in 1957 - as I said before, there were ferries between Havana and Florida, and day trips and weekend trips, from the US to Cuba, and from Cuba to the US, were a common, everyday occurrence.
What was necessary was a visa to enter the US if you were coming to live here - and that was granted by the US government, not the “Batista regime”.