Any ideas how I can install Net::Stripe given the compiler limitation?

You'd have to perform the download (e.g. cpanm --look Date::Simple), perl Makefile.PL and make steps of the installation on a sufficiently-similar machine. Then you'd copy the build directory over to the target machine and perfrom the make test and make install steps there.

I would first try to install the C compiler (from binaries) on the target machine.