Just a wild guess, but one conceivable reason that $::version would not work is that it's being used before it has been assigned its value. After all - as you have use Arithmetic; - the code is in an implicit BEGIN {...} block. In other words, in case there's any initialisation in the module, that code might be executed before $::version is ready.  For example, this would not work as one might expect at first:

#!/usr/bin/perl

use strict;
use warnings;

$::version = shift @ARGV;

BEGIN {  # emulate "use Arithmetic;"

  package Arithmetic;
  
  sub load_config {
      my $config = "/usr/local/myapp/$::version/properties";
      
      print "config: $config\n";
      # ...
  }
  
  load_config();
}
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In case you recognise any similarities to your scenario, you could try wrapping the assignment $::version = shift @ARGV; in a BEGIN block, which would make sure that things are being executed in the proper sequence.

Pass the variable into the constructor for the object that needs it then. Besides, a little refactoring up front to add such a facility rather than hacking it in is almost always worth while.