I wrote HTTP::Request::FromCurl, live at curl2lwp.psgi, which allows you to generate LWP::UserAgent code from Curl command lines. This could help with the transition to LWP::UserAgent.

But, as you already note, if the machine cannot install any SSL module (and you cannot build on a different but similar machine and copy the files from there), then you will not be able to make https connections from Perl at all without using an external program.