I could even get momentarily the necessary admin rights to install strawberry-there is no objection to having perl running on my corporate machine.

If you grab a "portable" edition of Strawberry Perl, you can just install it into your user directory - no admin rights required. That would be my recommendation.
Just download the perl-5.32.1 "Portable" edition (there's a choice of 32-bit or 64-bit, with most people opting for the latter) from https://strawberryperl.com/releases.html.
Unpack the tarball into some directory (let's call it "Strawberry") in your user space.
Then cd to that "Strawberry" folder and execute portableshell.bat. You're then set to go with strawberry perl.

Strawberry provides you with its own gmake.exe and gcc toolchain - and portableshell.bat has already prepended the location of those needed tools to your PATH environment variable.

You could then download the Tk source tarball to some location in your user directory, cd to that location and run 'cpan .' (without the quotes).
This means that the cpan utility does not have to perform the download (which was problematic for you).

If you're uncomfortable about using the gmake.exe provided by Strawberry (in c/bin) then let us know and we'll tell you how to get set up with dmake.exe (which is the make utlity that Strawberry used to use, and should still work fine.)
It's best if you could use that gmake.exe that Strawberry provides - with dmake.exe there might be a couple of hoops to jump through, though it should still be readily do-able.

Cheers,
Rob