Lots of great ways to do it - but this smells of an XY Problem to me. As in, why does package A inspect $self? I can imagine a few cases where this may be desirable (especially with dynamic subclasses - subclasses where the name is part of the data), although even then it's kinda suspect (just put the name in the $self hash).
In general, inheritance is done by simply putting overridable actions in their own subs, and then letting each package override the appropriate methods in the appropriate ways. No introspection required. Of course, you don't need to put them in separate files, either, but that's up to you.