You can make pseudo-private subs with references and my as follows. Pretty much the same as your first case but I think it looks a bit better ;)

#!perl
use strict;

package Whatever;

my $password = sub { 'passwd_here' };

sub reveal_pass {
    print "Authorised: ", $password->(), "\n";
}
  
package main;

Whatever::reveal_pass;

print "Unauthorised: ";
print ref $Whatever::password ? $Whatever::password->() : "no access!\
+n";

__END__
Authorised: passwd_here
Unauthorised: no access!
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