Just $0.02 but I believe you've found the end of the support for the HTTP abilities of the Extendables framework.
The 2 cent part is, although it's much more challenging, Adobe's script API can make use of other languages with more features, such as COM / VBScript on Windows and AppleScript on OSX. The script API can hand the request to those languages in addition (or a c++ library) and they can fully handle the request and then would need to return the data back to the script API. Not for the feint of heart.
Finally, while it's certainly not the most silky road, you might consider sort of a 'proxy', similar to how CORS is is sometimes handled. If you have a script on a web server (any language), say, PHP, you could send a HTTP request to it specifying the HTTPS target and any other arguments, while PHP then performs the HTTPS request and returns the information back to you in HTTP. Not elegant but there's always several ways to handle any situation.
We considered both of the options you describe and hoped there would be a better way to deal with this.
Anyways, thanks for your reply and we'll see how we'll fix this!
You're welcome and good luck!
Just a quick note: I've revisited the old GetURL and made a more modern version which does support https as well as http. All smoke and mirrors. More information and source code on my personal blog: