Maybe not reading it right, but I'd think you could just stash the config values in a class variable (so %my::base::_config or %my::branch_A::_config). You'd need something to walk up the @ISA to retrieve values if they're not present in the class of the instance (handwaving here and untested ):

sub _fetch { 
  my( $class, $key ) = @_; 
  no strict q{refs}; 
  my $opts = \%{ ref($class) . q{::_config} };
  if( exists $opts->{ $key } ) {
    return $opts->{ $key };
  } else {
    return SUPER->_fetch( $key );
  }
}

sub configure {
  my( $class, %opts ) = @_;
  no strict q{refs};
  my $opts = \%{ $class . q{::_config} };
  for my ($k, $v) ( each %opts ) {
    $opts->{ $k } = $v;
  }
  return;
}
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Edit: Adding stub configure