My first instinct would be to upgrade to a newer version of Perl. Not sure why you say that upgrading is not an option, but I won't
One option for you to do is to use App::FatPacker to "pack your dependencies onto your script file". I personally haven't used this module myself. For packing pure Perl modules needed by your script, there shouldn't be a problem. However, I'm not sure how well this module works with packing XS based modules.
Another option would be to use the pp utility from the PAR::Packer module to bundle your script into a stand-alone executable that can be run on another system - even if that system does not have Perl installed.
Since you're already familiar with Strawberry Perl and you can obviously transfer files to the system in question, here's another route that you could go. Download a portable version of Strawberry Perl. To use a version of portable Strawberry Perl, you just run a batch file that opens a new command prompt that has its environment variables set so that you can use the portable Strawberry Perl in just that command prompt. If you use a system with internet access, you can install any modules that you need to from CPAN. Then you can copy all of that plus your script to the system in question.