Please correct me, but how is this better than abs_path($0) ?

AFAIK is abs_path the problematic part, because the CWD might have changed after a chdir

I suppose that $0 is even better, since caller might only show an intermediate module instead of the root script (untested)

Please correct me, but how is this better than abs_path($0) ?

Because it answers the OP's question... "Is there a way to get a module to return the full system path of the script using the module?"

$ cat lib/Foo/Bar.pm 
package Foo::Bar;
use warnings;
use strict;
use Cwd qw/abs_path/;

sub foo {
    my $callerfile = (caller)[1];
    print "DEBUG: caller=<$callerfile> \$0=<$0>\n";
    return abs_path($callerfile);
}

1;

$ cat y.pl 
use warnings;
use strict;
use Foo::Bar;
print Foo::Bar::foo(), "\n";

$ cat x.pl 
use warnings;
use strict;
do './y.pl';

$ perl -Ilib x.pl 
DEBUG: caller=<./y.pl> $0=<x.pl>
/tmp/y.pl
[download]

Your emphasis the script using the module demonstrates the ambiguity of the OP's question. Is this about the script (which we usually identify as whatever is in $0) or is this the file from which a particular subroutine was called, as returned by caller? "Using a module" and "calling a subroutine in a module" are different things and can happen in different files.

Well semantics, but I see your point

"The script" for me is the root file executed.

YMMV

Cheers Rolf
_{(addicted to the 𐍀𐌴𐍂𐌻 Programming Language :)
Wikisyntax for the Monastery}