Ok, I got LWP via HTTPS working. I also have the XML to send. Next step...how do I put the SOAP wrapper around the XML payload and then send it via LWP?

Oh boy :) See Re: SOAP XML Request over HTTPS, look inside the modules like for example https://metacpan.org/source/IVANWILLS/W3C-SOAP-0.10/lib/W3C/SOAP/Client.pm

 $self->ua is LWP::UserAgent object ... or it can be a WWW::Mechanize (more convenient)

This is like low level SOAP, good luck to you :)