I don't think Ben was a cover-up name because, as others have rightly said, changing the first name and not the last is just pointless, they're bound to check you out if they're really looking.
I reckon Ben should have been his familiar name for several years before Luke's birth. Vader clearly never uses it in any of the OOT movies. Tarkin's obviously heard of Obi-Wan as well, but not as Ben. Obi-Wan fled to the outer rim and stayed there. They just never saw the point of going after him. The Republic was defeated and the Jedi Order decimated and scattered. I would suppose that the Empire expected the remaining Knights to die of old age and never be heard of again. They would also have been pretty confident that there weren't enough of them left and that those that were left were too old to regroup and start any trouble.