I can share my experience on cross-compiling of perl and it's modules, and my speach will rather prove it's not only unsupported, but rather quite impossible to do.

First of all, cross-compiler gcc on cygwin for Linux is very non-trivial, but still possible thing to do. But at second, you must have special Config.pm for those cross-purposes, and some more similar files, to do cross-compiling, and in no circumstances mix local Config.pm with cross one.

I sent about 10 patches to perl5-porters for 5.8.0-pre to do some initial steps of Cross-compiling when, while on Win32, you compile perl and it's modules for WinCE device.
I introduced an optionally-generated module Cross.pm which allows one to prepare makefile by running perl -MCross=wince Makefile.PL and then invoking of right "make" will cross-build with correct options.

While that approach could be developed further to cover your case, it is quite non-tested now, requires 5.8.0 anyway and is a way with a lot of tough places everywhere.

Which version of perl do they have on Linux?

As a last resort, I'll suggest you to write and run a simple shell or perl script that does similar to following:

cd /to/the/path/where-you-dloaded-module
perl Makefile.PL
make
make test
make install
[download]

Execute it somehow, and with some luck you'll succeed.

Best wishes,
Courage, the Cowardly Dog