You need to produce the stack trace yourself, since you are overriding the whole thing. Try:

 
#!/usr/bin/perl
use v5.14;
use Data::Dumper;

BEGIN {
    package Err;

    require Exporter;
    our @ISA = qw(Exporter);
    our @EXPORT = qw(err);

    use overload q("") => sub { my $e = shift; "ERROR $e->[0]: $e->[1]
+ \n".Carp::longmess()};

    sub new { my $class = shift; bless $_[0], $class }

    sub err { __PACKAGE__->new([@_]) }
}

package main;
BEGIN { Err->import }
use Carp;

sub f {
    g();
}

sub g {
    die err(403, "Permission denied");
}

f();
[download]

Which produces:

ERROR 403: Permission denied 
 at ./test.pl line 28.
    main::g() called at ./test.pl line 24
    main::f() called at ./test.pl line 31
[download]

UPDATE: I feel I should add that I don't see anything "more proper" in this way of handling stuff. Still, if you want to do it, have fun, and I hope my reply is of some use to you.

regards,
Luke Jefferson